SpaceGroup/0000755000175000017500000000000011153052500012112 5ustar segresegreSpaceGroup/t/0000755000175000017500000000000011153052500012355 5ustar segresegreSpaceGroup/t/test.t0000755000175000017500000000553011020062223013523 0ustar segresegre#!/usr/bin/perl -I../blib/lib/ use Test::Simple tests => 19; use Xray::SpaceGroup; ############################################################ ## Tests of extracting information from the database my $sg = Xray::SpaceGroup -> new({group=>'pc'}); ok($sg->group eq 'p c', "canonicalize space group"); ok($sg->given eq 'pc', "given group retained"); ok($sg->schoenflies eq 'c_s^2', "obtain schoenflies symbol"); ok($sg->class eq 'monoclinic', "obtain crystal class"); ok($sg->setting eq 'b_unique_1', "crystal setting"); my @positions = $sg->positions; my ($x, $y, $z) = (0.2, 0.3, 0.4); my @pos1 = (eval "$positions[1]->[0]", eval "$positions[1]->[1]", eval "$positions[1]->[2]" ); # ("$x", "-$y", "$z+1/2") ok($pos1[0] == 0.2 && $pos1[1] == -0.3 && $pos1[2] == 0.9, "compute from positions"); $sg = Xray::SpaceGroup -> new({group=>'f d -3 m'}); my @list = $sg->bravais; ok($#list == 8, "obtain Bravais translations"); $sg = Xray::SpaceGroup -> new({group=>'f d -3 m'}); @list = $sg->positions; ok($#list == 47, "obtain positions"); ############################################################ ## Tests of parsing space group symbols $sg = Xray::SpaceGroup -> new({group=>'pm3m'}); ok($sg->group eq 'p m -3 m', "no spaces in symbol"); $sg = Xray::SpaceGroup -> new({group=>'Pm3M'}); ok($sg->group eq 'p m -3 m', "mixed case in symbol"); $sg = Xray::SpaceGroup -> new({group=>'p m 3 m'}); ok($sg->group eq 'p m -3 m', "lots of spaces in symbol"); $sg = Xray::SpaceGroup -> new({group=>"pm\t3 \t m"}); ok($sg->group eq 'p m -3 m', "tabs in symbol"); $sg = Xray::SpaceGroup -> new({group=>'o_h^1'}); ok($sg->group eq 'p m -3 m', "shoenflies forward"); $sg = Xray::SpaceGroup -> new({group=>'o^1_h'}); ok($sg->group eq 'p m -3 m', "shoenflies backward"); $sg = Xray::SpaceGroup -> new({group=>'perovskite'}); ok($sg->group eq 'p m -3 m', "nickname"); $sg = Xray::SpaceGroup -> new({group=>221}); ok($sg->group eq 'p m -3 m', "number"); $sg = Xray::SpaceGroup -> new({group=>'p 4/m -3 2/m'}); ok($sg->group eq 'p m -3 m', "full symbol"); $sg = Xray::SpaceGroup -> new({group=>'p4/ m -3 2 /m'}); ok($sg->group eq 'p m -3 m', "full symbol, weird spacing"); $sg = Xray::SpaceGroup -> new({group=>'p 6_3 m c'}); ok($sg->group eq 'p 63 m c', "underscore for subscript"); SpaceGroup/lib/0000755000175000017500000000000011153052500012660 5ustar segresegreSpaceGroup/lib/Xray/0000755000175000017500000000000011153052500013603 5ustar segresegreSpaceGroup/lib/Xray/SpaceGroup.pm0000644000175000017500000007571411132742411016233 0ustar segresegrepackage Xray::SpaceGroup; =for Copyright . Copyright (c) 1999-2008 Bruce Ravel (bravel AT bnl DOT gov). All rights reserved. . This file is free software; you can redistribute it and/or modify it under the same terms as Perl itself. See The Perl Artistic License. . This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. =cut use strict; use warnings; #use diagnostics; use Carp; use Class::Std; use File::Spec; use List::MoreUtils qw(any true); use Regexp::List; use Storable; use Readonly; Readonly my $EPSILON => 0.00001; use vars qw($VERSION); use version; $VERSION = version->new("0.1.0"); # use Data::Dumper; sub identify_self { my @caller = caller; use File::Basename qw(dirname); return dirname($caller[1]); }; { my %params_of :ATTR; my $database = File::Spec->catfile(identify_self(), 'SpaceGroup', 'space_groups.db'); my $r_sg = retrieve($database); my $opt = Regexp::List->new; my $number_attr = $opt->list2re(qw(a b c alpha beta gamma)); my $sh_re = $opt->list2re(qw(hex hcp zincblende zns cubic salt perov perovskite gra graphite fcc salt nacl diamond bcc cscl)); sub START { my ($self, $ident, $arguments) = @_; #$self->set({file=>q{}, entries=>[]}); #print ref($arguments), $/; $arguments = {group=>$arguments} if (ref($arguments) eq 'SCALAR'); $self->set($arguments); }; sub set { my ($self, $arguments) = @_; foreach my $key (keys %$arguments) { my $k = lc $key; $params_of{ident $self}{$k} = $arguments->{$k}; if ($k eq 'group') { $params_of{ident $self}{given_group} = $arguments->{$k}; $params_of{ident $self}{setting} = 0; $self->_canonicalize_group; $self->_set_bravais; $self->_crystal_class; #$self->_determine_monoclinic; }; }; }; sub get { my ($self, @params) = @_; croak('$type: usage: get($key) or get(@keys)') if @_ < 2; my @values = (); foreach my $key (@params) { my $k = lc $key; push @values, $params_of{ident $self}{$k}; }; return wantarray ? @values : $values[0]; }; sub group :STRINGIFY { my ($self) = @_; return $params_of{ident $self}{space_group}; }; sub given { my ($self) = @_; return $params_of{ident $self}{given_group}; }; sub number :NUMERIFY { my ($self) = @_; return $r_sg->{$self}->{number}; }; sub full { my ($self) = @_; return $r_sg->{$self}->{full}; }; sub schoenflies { my ($self) = @_; return $r_sg->{$self}->{schoenflies}; }; sub thirtyfive { my ($self) = @_; return $r_sg->{$self}->{thirtyfive}; }; sub newsymbol { my ($self) = @_; return $r_sg->{$self}->{new_symbol}; }; sub class { my ($self) = @_; return $params_of{ident $self}{class}; }; sub nicknames { my ($self) = @_; my $list_ref = $r_sg->{$self}->{shorthand} || []; return @$list_ref; }; sub setting { my ($self) = @_; return $params_of{ident $self}{setting} }; sub bravais { my ($self) = @_; my $list_ref = $params_of{ident $self}{bravais} || []; return @$list_ref; }; sub warning { my ($self) = @_; return $params_of{ident $self}{warning}; }; sub positions { my ($self) = @_; my $list_ref = []; ## R groups in the rhombohedral setting if ( ($self->get('space_group') =~ m{\Ar}) and ($self->get('setting') eq 'rhombohedral') ) { $list_ref = $r_sg->{$self}->{rhombohedral}; ## monoclinic group settings } elsif ( $self->get('class') eq 'monoclinic' ) { my $this = $self->get('setting'); $list_ref = $r_sg->{$self}->{$this} || []; ## everything else uses the "positions" entry } else { $list_ref = $r_sg->{$self}->{positions} || []; }; return @$list_ref; }; sub report { my ($self) = @_; my $message = sprintf("Space group: %s (%d)\n", $self, $self); $message .= sprintf(" supplied symbol : %s\n", $self->given); $message .= sprintf(" crystal class : %s\n", $self->class); $message .= sprintf(" Schoenflies symbol : %s\n", $self->schoenflies); $message .= sprintf(" full symbol : %s\n", $self->full) if $self->full; $message .= sprintf(" 1935 symbol : %s\n", $self->thirtyfive) if $self->thirtyfive; $message .= sprintf(" new symbol : %s\n", $self->newsymbol) if $self->newsymbol; $message .= sprintf(" nicknames : %s\n", join(", ", $self->nicknames)) if $self->nicknames; $message .= sprintf(" crystal setting : %s\n", $self->setting); $message .= " Bravais translations :\n"; my @brav = map { _simple_fraction($_) } $self->bravais; $message .= " none\n" if not @brav; $message .= sprintf(" %-7s %-7s %-7s\n", @brav) if ($#brav == 2); $message .= sprintf(" %-7s %-7s %-7s\n %-7s %-7s %-7s\n", @brav) if ($#brav == 5); $message .= sprintf(" %-7s %-7s %-7s\n %-7s %-7s %-7s\n %-7s %-7s %-7s\n", @brav) if ($#brav == 8); $message .= " Positions :\n"; foreach my $p ($self->positions) { $message .= sprintf(" %-7s %-7s %-7s\n", @$p); }; return $message; }; sub _simple_fraction { # stringify Bravais fractions my ($val) = @_; return (abs($val - 1/2) < $EPSILON) ? '1/2' : (abs($val - 1/3) < $EPSILON) ? '1/3' : (abs($val - 2/3) < $EPSILON) ? '2/3' : '0'; }; sub _canonicalize_group { my ($self) = @_; my $symbol = $params_of{ident $self}{given_group}; my @mono3 = qw(b_unique c_unique a_unique); my @mono9 = qw(b_unique_1 b_unique_2 b_unique_3 c_unique_1 c_unique_2 c_unique_3 a_unique_1 a_unique_2 a_unique_3); # this is a null value $params_of{ident $self}{warning} = q{}; if (! $symbol) { $params_of{ident $self}{space_group} = q{}; $params_of{ident $self}{setting} = 0; $params_of{ident $self}{data} = q{}; $params_of{ident $self}{warning} = q{Your symbol could not be recognized as a space group symbol!}; return (q{},0); }; $symbol = lc($symbol); # lower case and ... $symbol =~ s{[!\#%*].*$}{}; # trim off comments $symbol =~ s{^\s+}{}; # trim leading spaces $symbol =~ s{\s+$}{}; # trim trailing spaces $symbol =~ s{\s+}{ }g; # ... single space $symbol =~ s{\s*/\s*}{/}g; # spaces around slash $symbol =~ s{2_1}{21}g; # replace `i 4_1' with `i 41' $symbol =~ s{3_([12])}{3$1}g; # and so on ... $symbol =~ s{4_([1-3])}{4$1}g; $symbol =~ s{6_([1-5])}{6$1}g; if ( ($symbol !~ m{[_^]}) and # schoen ($symbol !~ m{\A\d{1,3}\z}) and # 1-230 ($symbol !~ m{\A($sh_re)\z}io) # shorthands like 'cubic', 'zns' ) { #print $symbol; $symbol = _insert_spaces($symbol); #print "|$symbol|\n"; }; # this is the standard symbol if (exists($r_sg->{$symbol})) { $params_of{ident $self}{space_group} = $symbol; $params_of{ident $self}{setting} = (any {$r_sg->{$symbol}->{number} eq $_} (3..6, 10..12) ) ? "b_unique" : (any {$r_sg->{$symbol}->{number} eq $_} (7..9, 13..15) ) ? "b_unique_1" : "positions"; my $rhash = $r_sg->{$symbol}; $params_of{ident $self}{data} = $rhash; return ($symbol, 0); }; foreach my $sym (keys %$r_sg ) { next if ($sym eq "version"); my $rhash = $r_sg->{$sym}; # this is the Schoenflies symbol, (it # must have a caret in it) if ($symbol =~ /\^/) { $symbol =~ s/\s+//g; # no spaces $symbol =~ s/^v/d/g; # V -> D # put ^ and _ in correct order $symbol =~ s/([cdost])(\^[0-9]{1,2})(_[12346dihsv]{1,2})/$1$3$2/; if ((exists $r_sg->{$sym}->{schoenflies}) and ($symbol eq $r_sg->{$sym}->{schoenflies}) ) { $params_of{ident $self}{space_group} = $sym; $params_of{ident $self}{setting} = (any {$r_sg->{$sym}->{number} eq $_} (3..6, 10..12) ) ? "b_unique" : (any {$r_sg->{$sym}->{number} eq $_} (7..9, 13..15) ) ? "b_unique_1" : "positions"; $params_of{ident $self}{data} = $rhash; return ($sym, 0); }; }; # scalar valued fields # this is a number between 1 and 230 # or the 1935 symbol # or a double glide plane symbol # or the full symbol foreach my $field ("thirtyfive", "number", "new_symbol", "full") { if ( (exists $r_sg->{$sym}->{$field}) and ($symbol eq $r_sg->{$sym}->{$field}) ) { $params_of{ident $self}{space_group} = $sym; $params_of{ident $self}{setting} = (any {$r_sg->{$sym}->{number} eq $_} (3..6, 10..12) ) ? "b_unique" : (any {$r_sg->{$sym}->{number} eq $_} (7..9, 13..15) ) ? "b_unique_1" : "positions"; $params_of{ident $self}{data} = $rhash; return ($sym, 0); }; }; # now check the array values fields foreach my $field ("settings", "short", "shorthand") { if (exists($r_sg->{$sym}->{$field})) { my $i=0; my $count = -1; foreach my $setting ( @{$r_sg->{$sym}->{$field}} ) { ++$count; ++$i; my $s = ($field eq "settings") ? $i : 0; if ($symbol eq $setting) { $params_of{ident $self}{space_group} = $sym; if (any {$field eq $_} qw(settings short)) { if (any {$r_sg->{$sym}->{number} eq $_} (3..6, 10..12) ) { $params_of{ident $self}{setting} = $mono3[$count]; } elsif (any {$r_sg->{$sym}->{number} eq $_} (7..9, 13..15) ) { $params_of{ident $self}{setting} = $mono9[$count]; }; }; $params_of{ident $self}{data} = $rhash; return ($sym, $s); }; }; }; }; }; # this is not a symbol $params_of{ident $self}{space_group} = q{}; $params_of{ident $self}{setting} = 0; $params_of{ident $self}{data} = {}; $params_of{ident $self}{warning} = q{Your symbol could not be recognized as a space group symbol!}; return (q{},0); } ## This is the algorithm for dealing with user-supplied space group ## symbols that do not have the canonical single space separating the ## part of the symbol. sub _insert_spaces { my $sym = $_[0]; my ($first, $second, $third, $fourth) = ("", "", "", ""); ## a few groups don't follow the rules below ... ($sym =~ /\b([rhc])32\b/i) && return "$1 3 2"; ($sym =~ /\bp31([2cm])\b/i) && return "p 3 1 $1"; ($sym =~ /\bp(3[12]?)[22][12]\b/i) && return "p $1 2 1"; ($sym =~ /\bp(6[1-5]?)22\b/i) && return "p $1 2 2"; ($sym =~ /\b([fip])(4[1-3]?)32\b/i) && return "$1 $2 3 2"; ($sym =~ /\b([fipc])(4[1-3]?)(21?)(2)\b/i) && return "$1 $2 $3 $4"; ## the first symbol is always a single letter $first = substr($sym, 0, 1); my $index = 1; my $subsym = substr($sym, $index); if ($subsym =~ m{\A([ \t]+)}) { $index += length($1); }; if (substr($sym, $index, 4) =~ /([2346][12345]\/[mnabcd])/) { ## second symbol as in p 42/n c m $second = $1; $index += 4; } elsif (substr($sym, $index, 3) =~ /([2346]\/[mnabcd])/) { ## second symbol as in p 4/n n c $second = $1; $index += 3; } elsif (substr($sym, $index, 2) =~ /(-[1346])/) { ## second symbol as in p -3 1 m $second = $1; $index += 2; } elsif (substr($sym, $index, 2) =~ /(21|3[12]|4[123]|6[12345])/) { ## second symbol as in p 32 1 2 $second = $1; $index += 2; } else { $second = substr($sym, $index, 1); $index += 1; }; $subsym = substr($sym, $index); if ($subsym =~ m{\A([ \t]+)}) { $index += length($1); }; if (substr($sym, $index, 4) =~ /([2346][12345]\/[mnabcd])/) { ## third symbol as in full symbol p 21/c 21/c 2/n $third = $1; $index += 4; } elsif (substr($sym, $index, 3) =~ /([2346]\/[mnabcd])/) { ## third symbol as in full symbol p 4/m 21/b 2/m $third = $1; $index += 3; } elsif (substr($sym, $index, 2) =~ /(-[1346])/) { ## third symbol as in f d -3 m $third = $1; $index += 2; } elsif (substr($sym, $index, 2) =~ /(21|3[12]|4[123]|6[12345])/) { ## third symbol as in p 21 21 2 $third = $1; $index += 2; } else { $third = substr($sym, $index, 1); $index += 1; }; ($index < length($sym)) and $fourth = substr($sym, $index); $fourth =~ s/\A\s+//; $sym = join(" ", $first, $second, $third, $fourth); $sym =~ s/\s+$//; # trim trailing spaces return $sym; }; sub _set_bravais { my ($self) = @_; my %table = ( f => [ 0, 1/2, 1/2, 1/2, 0, 1/2, 1/2, 1/2, 0], i => [1/2, 1/2, 1/2], c => [1/2, 1/2, 0], a => [ 0, 1/2, 1/2], b => [1/2, 0, 1/2], r => [2/3, 1/3, 1/3, 1/3, 2/3, 2/3], ); my $group = $self->get("given_group"); my $g = lc(substr($group, 0, 1)); if ($g !~ m{[ficabr]}) { $group = $self->get("space_group"); $g = lc(substr($group, 0, 1)); }; my $setting = $self->get("setting"); $params_of{ident $self}{bravais} = []; $params_of{ident $self}{bravais} = $table{r} if (($g eq 'r') and ($setting eq "rhombohedral")); $params_of{ident $self}{bravais} = $table{$g} if ($g =~ m{[abcfi]}); return $self;; }; sub _crystal_class { my ($self) = @_; my $group = $self->get("space_group"); if (exists $r_sg->{$group}->{number}) { my $num = $r_sg->{$group}->{number}; my $class = ($num <= 0) ? q{} : ($num <= 2) ? "triclinic" : ($num <= 15) ? "monoclinic" : ($num <= 74) ? "orthorhombic" : ($num <= 142) ? "tetragonal" : ($num <= 167) ? "trigonal" : ($num <= 194) ? "hexagonal" : ($num <= 230) ? "cubic" : q{}; $params_of{ident $self}{class} = $class; if (($class eq 'monoclinic') and (not $params_of{ident $self}{setting})) { $params_of{ident $self}{setting} = "b_unique"; $params_of{ident $self}{setting} .= "_1" if any {$r_sg->{$self}->{number} == $_} (7,9,13,14,15); }; } else { $params_of{ident $self}{class} = q{}; }; return 1; }; sub fix { my ($self, $args) = @_; $args->{alpha} ||= 90; $args->{beta} ||= 90; $args->{gamma} ||= 90; ## determine correct setting for monoclinic groups by examining angles $self -> _determine_monoclinic($args); CHECK: { ($self->get('class') eq 'cubic') and do { if ( (abs($args->{a} - $args->{b}) > $EPSILON) or (abs($args->{b} - $args->{c}) > $EPSILON) or (abs($args->{a} - $args->{c}) > $EPSILON) or (abs($args->{alpha} - 90) > $EPSILON) or (abs($args->{beta} - 90) > $EPSILON) or (abs($args->{gamma} - 90) > $EPSILON) ) { $params_of{ident $self}{warning} = "Cubic space groups should have all axes equal and all angle of 90 degrees."; }; last CHECK; }; ($self->get('class') =~ m{(hexagonal|trigonal)}) and do { if ( (abs($args->{a} - $args->{b}) > $EPSILON) or (abs($args->{a} - $args->{c}) < $EPSILON) or (abs($args->{b} - $args->{c}) < $EPSILON) or (abs($args->{alpha} - 90) > $EPSILON) or (abs($args->{beta} - 90) > $EPSILON) or (abs($args->{gamma} - 120) > $EPSILON) ) { my $which = ucfirst($1); $params_of{ident $self}{warning} = "$which space groups should have a=b with c different, alpha=beta=90 degrees, and gamma=120 degrees."; }; last CHECK; }; ($self->get('class') eq 'tetragonal') and do { if ( (abs($args->{a} - $args->{b}) > $EPSILON) or (abs($args->{a} - $args->{c}) < $EPSILON) or (abs($args->{b} - $args->{c}) < $EPSILON) or (abs($args->{alpha} - 90) > $EPSILON) or (abs($args->{beta} - 90) > $EPSILON) or (abs($args->{gamma} - 90) > $EPSILON) ) { $params_of{ident $self}{warning} = "Tetragonal space groups should have a=b with c different, alpha=beta=gamma=90 degrees."; }; last CHECK; }; ($self->get('class') eq 'orthorhombic') and do { if ( (abs($args->{a} - $args->{b}) < $EPSILON) or (abs($args->{a} - $args->{c}) < $EPSILON) or (abs($args->{b} - $args->{c}) < $EPSILON) or (abs($args->{alpha} - 90) > $EPSILON) or (abs($args->{beta} - 90) > $EPSILON) or (abs($args->{gamma} - 90) > $EPSILON) ) { $params_of{ident $self}{warning} = "Orthorhombic space groups should have all axes unequal and alpha=beta=gamma=90 degrees."; }; last CHECK; }; ($self->get('class') eq 'monoclinic') and do { if ( (abs($args->{a} - $args->{b}) < $EPSILON) or (abs($args->{a} - $args->{c}) < $EPSILON) or (abs($args->{b} - $args->{c}) < $EPSILON) or ( (true {abs($_ - 90) > $EPSILON} ($args->{alpha}, $args->{beta}, $args->{gamma}) ) != 1) ) { $params_of{ident $self}{warning} = "Monoclinic space groups should have all axes unequal and one angle not equal to 90 degrees."; }; last CHECK; }; ($self->get('class') eq 'triclinic') and do { if ( (abs($args->{a} - $args->{b}) < $EPSILON) or (abs($args->{a} - $args->{c}) < $EPSILON) or (abs($args->{b} - $args->{c}) < $EPSILON) or (abs($args->{alpha} - 90) < $EPSILON) or (abs($args->{beta} - 90) < $EPSILON) or (abs($args->{gamma} - 90) < $EPSILON) ) { $params_of{ident $self}{warning} = "Triclinic space groups should have all axes unequal and all angles unequal."; }; last CHECK; }; }; }; sub _determine_monoclinic { my ($self, $args) = @_; $args->{alpha} ||= 90; $args->{beta} ||= 90; $args->{gamma} ||= 90; my ($group, $given, $class) = $self->get(qw(space_group given_group class)); return 0 if ($class ne "monoclinic"); ($given = $group) if ($given =~ m{\A\d+\z}); my $axis = ((abs( 90 - $args->{alpha} )) > $EPSILON) ? "a" : ((abs( 90 - $args->{beta} )) > $EPSILON) ? "b" : ((abs( 90 - $args->{gamma} )) > $EPSILON) ? "c" : q{}; #(! $axis) && do { # if ($self->get("angle")) { # $axis = lc(substr($self->get('angle'), 0, 1)); # $axis =~ tr/g/c/; # }; #}; #print "axis: $axis\n"; return 0 if (not $axis); # angles not set yet # if it has, then continue... my $setting = $axis . "_unique"; my $number = $r_sg->{$group}->{number}; ## these groups have one cell choice for each unique axis foreach my $n (3,4,5,6,8,10,11,12) { ($number == $n) && return $setting; }; ## groups 7, 13, 14 are p centered and have multiple cell choices #print "$group $given $axis\n"; if ($group =~ m{\Ap}) { if ($axis eq "b") { ($setting .= "_1") if ($given =~ m{c}i); ($setting .= "_2") if ($given =~ m{n}i); ($setting .= "_3") if ($given =~ m{a}i); } elsif ($axis eq "c") { ($setting .= "_1") if ($given =~ m{a}i); ($setting .= "_2") if ($given =~ m{n}i); ($setting .= "_3") if ($given =~ m{b}i); } elsif ($axis eq "a") { ($setting .= "_1") if ($given =~ m{b}i); ($setting .= "_2") if ($given =~ m{n}i); ($setting .= "_3") if ($given =~ m{c}i); }; }; ## groups 9, 15 are c centered and have multiple cell choices if ($group =~ m{\Ac}) { if ($axis eq "b") { ($setting .= "_1") if ($given =~ m{\Ac}i); ($setting .= "_2") if ($given =~ m{\Aa}i); ($setting .= "_3") if ($given =~ m{\Ai}i); } elsif ($axis eq "c") { ($setting .= "_1") if ($given =~ m{\Aa}i); ($setting .= "_2") if ($given =~ m{\Ab}i); ($setting .= "_3") if ($given =~ m{\Ai}i); } elsif ($axis eq "a") { ($setting .= "_1") if ($given =~ m{\Ab}i); ($setting .= "_2") if ($given =~ m{\Ac}i); ($setting .= "_3") if ($given =~ m{\Ai}i); }; }; ## if none of the preceding 6 blocks altered setting then there is a ## mismatch between the symbol and the unique axis, so return 0. ($setting = 0) if ($setting !~ /_[123]$/); $params_of{ident $self}{setting} = $setting; return $setting; }; }; 1; =head1 NAME Xray::SpaceGroup - Symmetry operations for the crystal space groups =head1 VERSION This documentation refers to libperlxray version 0.1. =head1 SYNOPSIS my $sg = Xray::SpaceGroup -> new("Pm3m"); $sg -> fix({a => $a, b => $b, c => $c, alpha => $alpha, beta => $beta, gamma => $gamma}) @symmetry_ops = $sg -> positions; @bravais_trans = $sg -> bravais; print $sg -> report; The two lists can be used to populate a unit cell, given a set of Wyckoff positions. The report is a user-readable summary of the properties of the space group. =head1 DESCRIPTION This provides an object-oriented interface to a database of space group symmetries transcribed from volume A of the International Tables for Crystallography. =head1 METHODS =head2 Accessor Methods =over 4 =item C This creates a new Xray::SpaceGroup object. my $sg = Xray::SpaceGroup -> new({group=>"Pm3m"}); The space group symbol can be a Hermann-Maguin symbol, a Schoenflies symbol, the number of the space group, one of several nicknames (diamond, cscl, etc), or a few other symbols from the International Tables. The H-M symbol can contain any number of spaces and is case insensitive. An overscore is indicated by a leading dash (C<->) and a subscript is simply written as a normal character, as in C

where "4 sub 2" is written as C<42>. A slash is indicated by a forward slash (C). The sub- and superscripts of the Schoenflies symbol can come in either order and are indicated by caret (C<^>) and underscore (C<_>). The nicknames are as follows: symbol number nicknames ------------------------------------------------------------ p 63 m c 186 graphite, gra p 63/m m c 194 hex, hcp f -4 3 m 216 zincblende, zns p m -3 m 221 cubic, cscl, perov, perovskite f m -3 m 225 fcc, salt, nacl f m -3 d 227 diamond i m -3 m 229 bcc =item C Set the crystal setting based on the values of the lattice constants and verify that the lattice constants make sense for the selected crystal class. $sg -> fix({a => $a, b => $b, c => $c, alpha => $alpha, beta => $beta, gamma => $gamma}) The first chore of this method is quite necessary for a space group with multiple settings. Selecting the correct setting requires knowledge of the lattice parameters. For many space groups, including all cubic groups, this is a no-op. =item C This will return a non-empty string if a problem was found in the C method. Most such problems result from a mismatch between the lattice constant values and what is expected for the crystal class. For instance, if a monoclinic space group symbol is given but all three angles are 90 degrees, a warning will be returned by this method. This sort of problem rarely requires that your program stops, so no explicit exception is thrown when C finds a problem. If you want to know whether a problem was found, you must explicitly call this method. =item C This method is the meaty part of this module. This returns a list of lists containing the symmetry operations for your space group and the appropriate crystal setting. Here is an example using a monoclinic space group, which has a short list of symmetry operations. use Data::Dumper; my $sg = Xray::SpaceGroup -> new({group=>'7'}); my @positions = $sg->positions; print Data::Dumper->Dump([\@positions], [qw(positions)]); ==prints==> $positions = [ [ '$x', '$y', '$z' ], [ '$x', '-$y', '$z+1/2' ] ]; The elements of these lists are strings and are intended to be evaluated using perl's eval function. For example, if you have a Wyckoff position of C<(0.2, 0.3, 0.4)>, then you might do this: ($x, $y, $z) = (0.2, 0.3, 0.4); @pos0 = (eval "$positions[0]->[0]", eval "$positions[0]->[1]", eval "$positions[0]->[2]" ); @pos1 = (eval "$positions[1]->[0]", eval "$positions[1]->[1]", eval "$positions[1]->[2]" ); This will result in C<@pos0 = (0.2, 0.3, 0.4)> and C<@pos2 = (0.2, -0.3, 0.9)>. You would, in practice, wrap these evaluations inside proper control structures. You might also use a L compartment if you are worried about the possibility of the database having been tainted. For high symmetry groups and high symmetry Wyckoff positions, these evaluations will generate the same positions repeatedly. It is the responsibility of your application to weed out these repetitions. =back There are also C and C methods, but properties of the space group should be obtained via the reporting methods listed below. =head2 Reporting Methods =over 4 =item C This writes a summary of the properties of the space group in a human-readable form. my $space = Xray::SpaceGroup -> new({group=>7}); print join(", ", $space->bravais), $/; ==prints==> Space group: p c (7) supplied symbol : 7 crystal class : monoclinic Schoenflies symbol : c_s^2 crystal setting : b_unique_1 Bravais translations : none Positions : $x $y $z $x -$y $z+1/2 =item C This returns the canonical space symbol for this space group. print $sg->group, $/; ==prints==> p m -3 m =item C This returns the space symbol supplied when the object was created. print $sg->given, $/; ==prints==> Pm3m =item C This returns the number of the space group as in the International Tables. print $sg->number, $/; ==prints==> 221 =item C This returns the full symbol of the space group as in the International Tables. print $sg->full, $/; ==prints==> p 4/m -3 2/m =item C This returns the Schoenflies symbol of the space group. print $sg->schoenflies, $/; ==prints==> o_h^1 =item C This returns the symbol of the space group as it appeared in the 1935 edition of the International Tables, if it was different from the canonical symbol. Otherwise this returns an empty string. print $sg->thirtyfive, $/; ==prints==> p m 3 m =item C This returns the new symbol of the space group as introduced by the IUCr nomenclature report of 1992. Only a handful of groups with glide planes have new symbols. An empty string is returned for most groups. my $sgnew = Xray::SpaceGroup -> new({group=>"a b a 2"}); print $sgnew->newsymbol, $/; ==prints==> a e a 2 =item C This returns the crystal class of the space group print $sg->class, $/; ==prints==> cubic =item C This returns the setting of the space group using the nomenclature of the database used by this module. If there is only one setting, this returns 0. For rhombohedral space groups, this returns a string -- either "positions" or "rhombohedral" -- indicating which set of symmetry operations should be used. For most monoclinic groups, this returns one of "b_unique", "c_unique", or "a_unique", indicating which set of symmetry operations should be used. If the beta angle is not 90 degrees, the "b_unique" setting should be used. If the gamma or alpha angles are not 90 degrees, the "c_unique" or "a_unique" settings should be used, respectively. For several monoclinic space groups, there are additional settings for each unique axis. These are indicated as "b_unique_1", "b_unique_2", "b_unique_3", and so on. =item C This returns a 0, 3, 6, or 9 element list containing the Bravais translation vectors associated with the space group. my $diamond = Xray::SpaceGroup -> new({group=>"f d -3 m"}); print join(", ", $diamond->bravais), $/; ==prints==> 0.000, 0.500, 0.500, 0.500, 0.000, 0.500, 0.500, 0.500, 0.000 Each triplet is a Bravais translation. The Bravais translations attempt to be sensitive to the specified crystal setting. If you use ambiguous input (i.e. the number or the Schoenflies symbol) it is possible that a Bravais translation other than the one you want will be returned. The telepathic interface is planned for version 2.0. =back =head1 COERCIONS When the reference to the Xray::SpaceGroup object is used in string context, the C method is returned. When used in numerical context, the C method is returned. =head1 CONFIGURATION AND ENVIRONMENT This requires that the F file, which is generated by the F script, be installed in the correct location. There are no other configuration options. =head1 DEPENDENCIES This module uses several things from the standard distribution along with: L L L L =head1 BUGS AND LIMITATIONS Missing features: =over 4 =item * Groups 5, 8, 12 list 9 symbols, but 3 sets of positions. What's up with that? =item * Groups 12 - 15, short notation is ambiguous, requires angles to resolve. =item * C<_determine_monoclinic> should use alpha/beta/gamma, C<_canonicalize_group> already parsed the given_group =item * Recognize setting for R groups =item * Rotate symmetry ops for orthorhombic groups to the setting specified by the symbol. In atoms, I rotate the coordinates and rotate them back. Rotating the symmetry ops is a better, more general purpose solution. =item * Handle alternate tetragonal group settings here rather than in the application. =back Please report problems to Bruce Ravel (bravel AT bnl DOT gov) Patches are welcome. =head1 AUTHOR Bruce Ravel (bravel AT bnl DOT gov) http://cars9.uchicago.edu/~ravel/software/ =head1 ACKNOWLEDGEMENTS Julie Cross and Matt Newville gave me a copy of volume A of the International Tables for Crystallography as a graduation present from grad school. Was that a blessing or a curse? Saulius Grazulis, whose useful feedback inspired this most recent rewrite of this code. Earlier versions of Atoms benefited from the help and criticism of Shelly Kelly, Daniel Haskel, Chuck Bouldin, Hannes Fischer, Glenn Forney, Chris Glover, John Rehr, Hubert Renevier, Julia Wienold, Andrzej Wilamowski, Boyan Boyanovich, Ed Stern, Hans Stragier, Kalle Voss, Steve Zabinsky, and Yanjun Zhang. All the users of my codes over the years have driven me to provide the best possible code. =head1 LICENCE AND COPYRIGHT Copyright (c) 1999-2008 Bruce Ravel (bravel AT bnl DOT gov). All rights reserved. This module is free software; you can redistribute it and/or modify it under the same terms as Perl itself. See L. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. =cut SpaceGroup/data/0000755000175000017500000000000011153052500013023 5ustar segresegreSpaceGroup/data/space_groups.db.PL0000755000175000017500000040261411020062223016344 0ustar segresegre#! /usr/bin/perl -w ###################################################################### ## This program is copyright (c) 1998, 1999, 2001, 2007, 2008 Bruce Ravel ## ## http://feff.phys.washington.edu/~ravel/software/atoms/ ## ## ------------------------------------------------------------------- ## All rights reserved. This program is free software; you can ## redistribute it and/or modify it under the same terms as Perl ## itself. ## ## This program is distributed in the hope that it will be useful, ## but WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## Artistic License for more details. ## ------------------------------------------------------------------- ###################################################################### ## Time-stamp: <2008-05-30 14:39:46 bruce> ###################################################################### ## ## This program write the space groups data to a database file using ## perl's Storable format. The database filename space_groups.db. ## It would be a simple matter to target some other database format ## for the output. An example using a multilevel database is ## commented out. ## ## This program is a plain text representation of the space groups ## data base. The purpose of this program is to store that data in a ## form that is faster in operation and language agnostic. ## ## See Xray::Crystal::Cell in demeter for details about using the ## contents of this database. ## ## I intend that this program be run once at install time to create ## the database. After that it should only be necessary to rerun ## this program if the database in this file is updated. ## ## Each entry of %space_groups looks rather like this: ## "s y/m bo l" => { ## number => 712, number in table ## schoenflies => "a_b^c", Schoenflies notation ## full => "", full Hermann-Maguin notation ## settings => [ ba-c, cab, -cba, bca, a-cb], lattice permutations ## shiftvec => [ ], vector for groups with two origins in ITC ## new_symbol => "", double glide plane symbol, cf. sec 1.3 ## thirtyfive => "", symbols from the 1935 edition ## positions => [ internal symmetries ## [$x, $y, $z], ## ], ## rhombohedral=> [ internal symmetries, rhombohedral ## [$x, $y, $z], ## ], ## }, ## ## The symmetry operations in the positions entry are in the form of ## a list of lists. Each list in the LoL is a list of three strings. ## The strings are intended to be eval-ed to perform the symmetry ## operations. ## ###################################################################### ## Code: use File::Spec; ##Storable: use Storable qw/nstore/; ##end Storable: ##MLDBM: ## use MLDBM qw(DB_File Storable); ## use Fcntl; ##end MLDBM: my $version = 1.8; $| = 1; print "Space group database generation tool $version$/"; print " Munching on space group database.\n"; my %space_groups = ( 'version' => $version, ## triclinic groups 'p 1' => { number => 1, schoenflies => 'c_1^1', positions => [['$x', '$y', '$z']] }, 'p -1' => { number => 2, schoenflies => 'c_i^1', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '-$z']] }, ## monoclinic groups 'p 2' => { number => 3, schoenflies => 'c_2^1', settings => ['p 1 2 1', 'p 1 1 2', 'p 2 1 1'], b_unique => [['$x', '$y', '$z'], ['-$x', '$y', '-$z']], c_unique => [['$x', '$y', '$z'], ['-$x', '-$y', '$z']], a_unique => [['$x', '$y', '$z'], ['$x', '-$y', '-$z']] }, 'p 21' => { number => 4, schoenflies => 'c_2^2', settings => ['p 1 21 1', 'p 1 1 21', 'p 21 1 1'], b_unique => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '-$z']], c_unique => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2']], a_unique => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '-$z']] }, 'c 2' => { number => 5, schoenflies => 'c_2^3', short => ['c 2', 'a 2', 'i 2', 'a 2', 'b 2', 'i 2', 'b 2', 'c 2', 'i 2'], settings => ['c 1 2 1', 'a 1 2 1', 'i 1 2 1', 'a 1 1 2', 'b 1 1 2', 'i 1 1 2', 'b 2 1 1', 'c 2 1 1', 'i 2 1 1'], b_unique => [['$x', '$y', '$z'], ['-$x', '$y', '-$z']], c_unique => [['$x', '$y', '$z'], ['-$x', '-$y', '$z']], a_unique => [['$x', '$y', '$z'], ['$x', '-$y', '-$z']] }, 'p m' => { number => 6, schoenflies => 'c_s^1', settings => ['p 1 m 1', 'p 1 1 m', 'p m 1 1'], b_unique => [['$x', '$y', '$z'], ['$x', '-$y', '$z']], c_unique => [['$x', '$y', '$z'], ['$x', '$y', '-$z']], a_unique => [['$x', '$y', '$z'], ['-$x', '$y', '$z']] }, 'p c' => { number => 7, schoenflies => 'c_s^2', # note that the short notation is ambiguous short => ['p c', 'p n', 'p a', 'p a', 'p n', 'p b', 'p b', 'p n', 'p c'], settings => ['p 1 c 1', 'p 1 n 1', 'p 1 a 1', 'p 1 1 a', 'p 1 1 n', 'p 1 1 b', 'p b 1 1', 'p n 1 1', 'p c 1 1'], b_unique_1 => [['$x', '$y', '$z'], ['$x', '-$y', '$z+1/2']], b_unique_2 => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '$z+1/2']], b_unique_3 => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '$z']], c_unique_1 => [['$x', '$y', '$z'], ['$x+1/2', '$y', '-$z']], c_unique_2 => [['$x', '$y', '$z'], ['$x+1/2', '$y+1/2', '-$z']], c_unique_3 => [['$x', '$y', '$z'], ['$x', '$y+1/2', '-$z']], a_unique_1 => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '$z']], a_unique_2 => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '$z+1/2']], a_unique_3 => [['$x', '$y', '$z'], ['-$x', '$y', '$z+1/2']] }, 'c m' => { number => 8, schoenflies => 'c_s^3', # note that the short notation is ambiguous short => ['c m', 'a m', 'i m', 'a m', 'b m', 'i m', 'b m', 'c m', 'i m'], settings => ['c 1 m 1', 'a 1 m 1', 'i 1 m 1', 'a 1 1 m', 'b 1 1 m', 'i 1 1 m', 'b m 1 1', 'c m 1 1', 'i m 1 1'], b_unique => [['$x', '$y', '$z'], ['$x', '-$y', '$z']], c_unique => [['$x', '$y', '$z'], ['$x', '$y', '-$z']], a_unique => [['$x', '$y', '$z'], ['-$x', '$y', '$z']] }, 'c c' => { number => 9, schoenflies => 'c_s^4', short => ['c c', 'a n', 'i a', 'a a', 'b n', 'i b', 'b b', 'c n', 'i c'], settings => ['c 1 c 1', 'a 1 n 1', 'i 1 a 1', 'a 1 1 a', 'b 1 1 n', 'i 1 1 b', 'b b 1 1', 'c n 1 1', 'i c 1 1'], b_unique_1 => [['$x', '$y', '$z'], ['$x', '-$y', '$z+1/2']], b_unique_2 => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '$z+1/2']], b_unique_3 => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '$z']], c_unique_1 => [['$x', '$y', '$z'], ['$x+1/2', '$y', '-$z']], c_unique_2 => [['$x', '$y', '$z'], ['$x+1/2', '$y+1/2', '-$z']], c_unique_3 => [['$x', '$y', '$z'], ['$x', '$y+1/2', '-$z']], a_unique_1 => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '$z']], a_unique_2 => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '$z+1/2']], a_unique_3 => [['$x', '$y', '$z'], ['-$x', '$y', '$z+1/2']] }, 'p 2/m' => { number => 10, schoenflies => 'c_2h^1', settings => ['p 1 2/m 1', 'p 1 1 2/m', 'p 2/m 1 1'], b_unique => [['$x', '$y', '$z'], ['-$x', '$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '-$y', '$z']], c_unique => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z']], a_unique => [['$x', '$y', '$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['-$x', '$y', '$z']] }, 'p 21/m' => { number => 11, schoenflies => 'c_2h^2', settings => ['p 1 21/m 1', 'p 1 1 21/m', 'p 21/m 1 1'], b_unique => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '-$y+1/2', '$z']], c_unique => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z+1/2']], a_unique => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['-$x+1/2', '$y', '$z']] }, 'c 2/m' => { number => 12, schoenflies => 'c_2h^3', # note that the short notation is ambiguous short => ['c 2/m', 'a 2/m', 'i 2/m', 'a 2/m', 'b 2/m', 'i 2/m', 'b 2/m', 'c 2/m', 'i 2/m'], settings => ['c 1 2/m 1', 'a 1 2/m 1', 'i 1 2/m 1', 'a 1 1 2/m', 'b 1 1 2/m', 'i 1 1 2/m', 'b 2/m 1 1', 'c 2/m 1 1', 'i 2/m 1 1'], b_unique => [['$x', '$y', '$z'], ['-$x', '$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '-$y', '$z']], c_unique => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z']], a_unique => [['$x', '$y', '$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['-$x', '$y', '$z']] }, 'p 2/c' => { number => 13, schoenflies => 'c_2h^4', # note that the short notation is ambiguous short => ['p 2/c', 'p 2/n', 'p 2/a', 'p 2/a', 'p 2/n', 'p 2/b', 'p 2/b', 'p 2/n', 'p 2/c'], settings => ['p 1 2/c 1', 'p 1 2/n 1', 'p 1 2/a 1', 'p 1 1 2/a', 'p 1 1 2/n', 'p 1 1 2/b', 'p 2/a 1 1', 'p 2/n 1 1', 'p 2/c 1 1'], b_unique_1 => [['$x', '$y', '$z'], ['-$x', '$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '-$y', '$z+1/2']], b_unique_2 => [['$x', '$y', '$z'], ['-$x+1/2', '$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '-$y', '$z+1/2']], b_unique_3 => [['$x', '$y', '$z'], ['-$x+1/2', '$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '-$y', '$z']], c_unique_1 => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z']], c_unique_2 => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z']], c_unique_3 => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z'], ['-$x', '-$y', '-$z'], ['$x', '$y+1/2', '-$z']], a_unique_1 => [['$x', '$y', '$z'], ['$x', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['-$x', '$y+1/2', '$z']], a_unique_2 => [['$x', '$y', '$z'], ['$x', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['-$x', '$y+1/2', '$z+1/2']], a_unique_3 => [['$x', '$y', '$z'], ['$x', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['-$x', '$y', '$z+1/2']] }, 'p 21/c' => { number => 14, schoenflies => 'c_2h^5', # note that the short notation is ambiguous short => ['p 21/c', 'p 21/n', 'p 21/a', 'p 21/a', 'p 21/n', 'p 21/b', 'p 21/b', 'p 21/n', 'p 21/c'], settings => ['p 1 21/c 1', 'p 1 21/n 1', 'p 1 21/a 1', 'p 1 1 21/a', 'p 1 1 21/n', 'p 1 1 21/b', 'p 21/b 1 1', 'p 21/n 1 1', 'p 21/c 1 1'], b_unique_1 => [['$x', '$y', '$z'], ['-$x', '$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '-$y+1/2', '$z+1/2']], b_unique_2 => [['$x', '$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '-$y+1/2', '$z+1/2']], b_unique_3 => [['$x', '$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '-$y+1/2', '$z']], c_unique_1 => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2']], c_unique_2 => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z+1/2']], c_unique_3 => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y+1/2', '-$z+1/2']], a_unique_1 => [['$x', '$y', '$z'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['-$x+1/2', '$y+1/2', '$z']], a_unique_2 => [['$x', '$y', '$z'], ['$x+1/2', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['-$x+1/2', '$y+1/2', '$z+1/2']], a_unique_3 => [['$x', '$y', '$z'], ['$x+1/2', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['-$x+1/2', '$y', '$z+1/2']] }, 'c 2/c' => { number => 15, schoenflies => 'c_2h^6', # note that the short notation is ambiguous short => ['c 2/c', 'a 2/n', 'i 2/a', 'a 2/a', 'b 2/n', 'i 2/b', 'b 2/b', 'c 2/n', 'i 2/c'], settings => ['c 1 2/c 1', 'a 1 2/n 1', 'i 1 2/a 1', 'a 1 1 2/a', 'b 1 1 2/n', 'i 1 1 2/b', 'b 2/a 1 1', 'c 2/n 1 1', 'i 2/c 1 1'], b_unique_1 => [['$x', '$y', '$z'], ['-$x', '$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '-$y', '$z+1/2']], b_unique_2 => [['$x', '$y', '$z'], ['-$x+1/2', '$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '-$y', '$z+1/2']], b_unique_3 => [['$x', '$y', '$z'], ['-$x+1/2', '$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '-$y', '$z']], c_unique_1 => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z']], c_unique_2 => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z']], c_unique_3 => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z'], ['-$x', '-$y', '-$z'], ['$x', '$y+1/2', '-$z']], a_unique_1 => [['$x', '$y', '$z'], ['$x', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['-$x', '$y+1/2', '$z']], a_unique_2 => [['$x', '$y', '$z'], ['$x', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['-$x', '$y+1/2', '$z+1/2']], a_unique_3 => [['$x', '$y', '$z'], ['$x', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['-$x', '$y', '$z+1/2']], }, ## orthorhombic groups 'p 2 2 2' => { number => 16, schoenflies => 'd_2^1', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z']], }, 'p 2 2 21' => { number => 17, schoenflies => 'd_2^2', settings => [ 'p 2 2 21', 'p 21 2 2', 'p 21 2 2', 'p 2 21 2', 'p 2 21 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z']], }, 'p 21 21 2' => { number => 18, schoenflies => 'd_2^3', settings => [ 'p 21 21 2', 'p 2 21 21', 'p 2 21 21', 'p 21 2 21', 'p 21 2 21'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z']], }, 'p 21 21 21' => { number => 19, schoenflies => 'd_2^4', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z']], }, 'c 2 2 21' => { number => 20, schoenflies => 'd_2^5', settings => [ 'c 2 2 21', 'a 21 2 2', 'a 21 2 2', 'b 2 21 2', 'b 2 21 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z']], }, 'c 2 2 2' => { number => 21, schoenflies => 'd_2^6', settings => ['c 2 2 2', 'a 2 2 2', 'a 2 2 2', 'b 2 2 2', 'b 2 2 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z']], }, 'f 2 2 2' => { number => 22, schoenflies => 'd_2^7', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z']], }, 'i 2 2 2' => { number => 23, schoenflies => 'd_2^8', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z']], }, 'i 21 21 21' => { number => 24, schoenflies => 'd_2^9', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z']], }, 'p m m 2' => { number => 25, schoenflies => 'c_2v^1', settings => ['p m m 2', 'p 2 m m', 'p 2 m m', 'p m 2 m', 'p m 2 m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'p m c 21' => { number => 26, schoenflies => 'c_2v^2', settings => ['p c m 21', 'p 21 m a', 'p 21 a m', 'p b 21 m', 'p m 21 b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z']], }, 'p c c 2' => { number => 27, schoenflies => 'c_2v^3', settings => ['p c c 2', 'p 2 a a', 'p 2 a a', 'p b 2 b', 'p b 2 b' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2']], }, 'p m a 2' => { number => 28, schoenflies => 'c_2v^4', settings => ['p m a 2', 'p b m 2', 'p 2 m b', 'p 2 c m', 'p m 2 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y', '$z'], ['-$x+1/2', '$y', '$z']], }, 'p c a 21' => { number => 29, schoenflies => 'c_2v^5', settings => ['p b c 21', 'p 21 a b', 'p 21 c a', 'p c 21 b', 'p b 21 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['$x+1/2', '-$y', '$z'], ['-$x+1/2', '$y', '$z+1/2']], }, 'p n c 2' => { number => 30, schoenflies => 'c_2v^6', settings => ['p c n 2', 'p 2 n a', 'p 2 a n', 'p b 2 n', 'p n 2 b' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2']], }, 'p m n 21' => { number => 31, schoenflies => 'c_2v^7', settings => ['p n m 21', 'p 21 m n', 'p 21 n m', 'p n 21 m', 'p m 21 n' ], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y', '$z']], }, 'p b a 2' => { number => 32, schoenflies => 'c_2v^8', settings => ['p b a 2', 'p 2 c b', 'p 2 c b', 'p c 2 a', 'p c 2 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z']], }, 'p n a 21' => { number => 33, schoenflies => 'c_2v^9', settings => ['p b n 21', 'p 21 n b', 'p 21 c n', 'p c 21 n', 'p n 21 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z+1/2']], }, 'p n n 2' => { number => 34, schoenflies => 'c_2v^10', settings => ['p n n 2', 'p 2 n n', 'p 2 n n', 'p n 2 n', 'p n 2 n' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z+1/2']], }, 'c m m 2' => { number => 35, schoenflies => 'c_2v^11', settings => ['c m m 2', 'a 2 m m', 'a 2 m m', 'b m 2 m', 'b m 2 m' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'c m c 21' => { number => 36, schoenflies => 'c_2v^12', settings => ['c c m 21', 'a 21 m a', 'a 21 a m', 'b b 21 m', 'b m 21 b' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z']], }, 'c c c 2' => { number => 37, schoenflies => 'c_2v^13', settings => ['c c c 2', 'a 2 a a', 'a 2 a a', 'b b 2 b', 'b b 2 b' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2']], }, 'a m m 2' => { number => 38, schoenflies => 'c_2v^14', settings => ['b m m 2', 'b 2 m m', 'c 2 m m', 'c m 2 m', 'a m 2 m' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'a b m 2' => { number => 39, schoenflies => 'c_2v^15', settings => ['b m a 2', 'b 2 c m', 'c 2 m b', 'c m 2 a', 'a c 2 m' ], new_symbol => 'a e m 2', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y+1/2', '$z'], ['-$x', '$y+1/2', '$z']], }, 'a m a 2' => { number => 40, schoenflies => 'c_2v^16', settings => ['b b m 2', 'b 2 m b', 'c 2 c m', 'c c m 2', 'a m 2 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y', '$z'], ['-$x+1/2', '$y', '$z']], }, 'a b a 2' => { number => 41, schoenflies => 'c_2v^17', settings => ['b b a 2', 'b 2 c b', 'c 2 c b', 'c c 2 a', 'a c 2 a' ], new_symbol => 'a e a 2', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z']], }, 'f m m 2' => { number => 42, schoenflies => 'c_2v^18', settings => ['f m m 2', 'f 2 m m', 'f 2 m m', 'f m 2 m', 'f m 2 m' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'f d d 2' => { number => 43, schoenflies => 'c_2v^19', settings => ['f d d 2', 'f 2 d d', 'f 2 d d', 'f d 2 d', 'f d 2 d' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/4', '-$y+1/4', '$z+1/4'], ['-$x+1/4', '$y+1/4', '$z+1/4']], }, 'i m m 2' => { number => 44, schoenflies => 'c_2v^20', settings => ['i m m 2', 'i 2 m m', 'i 2 m m', 'i m 2 m', 'i m 2 m' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'i b a 2' => { number => 45, schoenflies => 'c_2v^21', settings => ['i b a 2', 'i 2 c b', 'i 2 c b', 'i c 2 a', 'i c 2 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z']], }, 'i m a 2' => { number => 46, schoenflies => 'c_2v^22', settings => ['i b m 2', 'i 2 m b', 'i 2 c m', 'i c 2 m', 'i m 2 a' ], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$x+1/2', '-$y', '$z'], ['-$x+1/2', '$y', '$z']], }, 'p m m m' => { number => 47, schoenflies => 'd_2h^1', full => 'p 2/m 2/m 2/m', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'p n n n' => { number => 48, schoenflies => 'd_2h^2', full => 'p 2/n 2/n 2/n', shiftvec => [1/4, 1/4, 1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2']], }, 'p c c m' => { number => 49, schoenflies => 'd_2h^3', full => 'p 2/c 2/c 2/m', settings => ['p c c m', 'p m a a', 'p m a a', 'p b m b', 'p b m b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2']], }, 'p b a n' => { number => 50, schoenflies => 'd_2h^4', full => 'p 2/b 2/a 2/n', settings => ['p b a n', 'p n c b', 'p n c b', 'p c n a', 'p c n a'], shiftvec => [1/4, 1/4, 0], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '-$z'], ['$x', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '$z'], ['-$x', '$y+1/2', '$z']], }, 'p m m a' => { number => 51, schoenflies => 'd_2h^5', full => 'p 21/m 2/m 2/a', settings => ['p m m b', 'p b m m', 'p c m m', 'p m c m', 'p m a m'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x+1/2', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x+1/2', '$y', '$z']], }, 'p n n a' => { number => 52, schoenflies => 'd_2h^6', full => 'p 2/n 21/n 2/a', settings => ['p n n b', 'p b n n', 'p c n n', 'p n c n', 'p n a n'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z+1/2'], ['$x', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2']], }, 'p m n a' => { number => 53, schoenflies => 'd_2h^7', full => 'p 2/m 2/n 21/a', settings => ['p n m b', 'p b m n', 'p c n m', 'p n c m', 'p m a n'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y', '$z']], }, 'p c c a' => { number => 54, schoenflies => 'd_2h^8', full => 'p 21/c 2/c 2/a', settings => ['p c c b', 'p b a a', 'p c a a', 'p b c b', 'p b a b'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2']], }, 'p b a m' => { number => 55, schoenflies => 'd_2h^9', full => 'p 21/b 21/a 2/m', settings => ['p b a m', 'p m c b', 'p m c b', 'p c m a', 'p c m a'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z']], }, 'p c c n' => { number => 56, schoenflies => 'd_2h^10', full => 'p 21/c 21/c 2/n', settings => ['p c c n', 'p n a a', 'p n a a', 'p b n b', 'p b n b'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y', '$z+1/2']], }, 'p b c m' => { number => 57, schoenflies => 'd_2h^11', full => 'p 2/b 21/c 21/m', settings => ['p c a m', 'p m c a', 'p m a b', 'p b m a', 'p c m b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x', '$y+1/2', '$z']], }, 'p n n m' => { number => 58, schoenflies => 'd_2h^12', full => 'p 21/n 21/n 2/m', settings => ['p n n m', 'p m n n', 'p m n n', 'p n m n', 'p n m n'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z+1/2']], }, 'p m m n' => { number => 59, schoenflies => 'd_2h^13', full => 'p 21/m 21/m 2/n', shiftvec => [1/4, 1/4, 0], settings => ['p m m n', 'p n m m', 'p n m m', 'p m n m', 'p m n m'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '$z']], }, 'p b c n' => { number => 60, schoenflies => 'd_2h^14', full => 'p 21/b 2/c 21/n', settings => ['p c a n', 'p n c a', 'p n a b', 'p b n a', 'p c n b'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x', '$y', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z+1/2'], ['$x', '-$y', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z']], }, 'p b c a' => { number => 61, schoenflies => 'd_2h^15', full => 'p 21/b 21/c 21/a', settings => ['p c a b', 'p b c a', 'p c a b', 'p b c a', 'p c a b'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z']], }, 'p n m a' => { number => 62, schoenflies => 'd_2h^16', full => 'p 21/n 21/m 21/a', settings => ['p m n b', 'p b n m', 'p c m n', 'p m c n', 'p n a m'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z+1/2']], }, 'c m c m' => { number => 63, schoenflies => 'd_2h^17', full => 'c 2/m 2/c 21/m', settings => ['c c m m', 'a m m a', 'a m a m', 'b b m m', 'b m m b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z+1/2'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z']], }, 'c m c a' => { number => 64, schoenflies => 'd_2h^18', full => 'c 2/m 2/c 21/a', new_symbol => 'c m c e', settings => ['c c m b', 'a b m a', 'a c a m', 'b b c m', 'b m a b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y+1/2', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x', '$y', '$z']], }, 'c m m m' => { number => 65, schoenflies => 'd_2h^19', full => 'c 2/m 2/m 2/m', settings => ['c m m m', 'a m m m', 'a m m m', 'b m m m', 'b m m m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'c c c m' => { number => 66, schoenflies => 'd_2h^20', full => 'c 2/c 2/c 2/m', settings => ['c c c m', 'a m a a', 'a m a a', 'b b m b', 'b b m b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2']], }, 'c m m a' => { number => 67, schoenflies => 'd_2h^21', full => 'c 2/m 2/m 2/a', new_symbol => 'c m m e', settings => ['c m m b', 'a b m m', 'a c m m', 'b m c m', 'b m a m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z'], ['-$x', '$y+1/2', '-$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y+1/2', '-$z'], ['$x', '-$y+1/2', '$z'], ['-$x', '$y', '$z']], }, 'c c c a' => { number => 68, schoenflies => 'd_2h^22', full => 'c 2/c 2/c 2/a', new_symbol => 'c c c e', shiftvec => [0, 1/4, 1/4], settings => ['c c c b', 'a b a a', 'a c a a', 'b b c b', 'b b a b'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z'], ['-$x', '$y', '-$z+1/2'], ['$x+1/2', '-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x+1/2', '$y', '$z+1/2']], }, 'f m m m' => { number => 69, schoenflies => 'd_2h^23', full => 'f 2/m 2/m 2/m', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'f d d d' => { number => 70, schoenflies => 'd_2h^24', full => 'f 2/d 2/d 2/d', shiftvec => [-1/8, -1/8, -1/8], positions => [['$x', '$y', '$z'], ['-$x+3/4', '-$y+3/4', '$z'], ['-$x+3/4', '$y', '-$z+3/4'], ['$x', '-$y+3/4', '-$z+3/4'], ['-$x', '-$y', '-$z'], ['$x+1/4', '$y+1/4', '-$z'], ['$x+1/4', '-$y', '$z+1/4'], ['-$x', '$y+1/4', '$z+1/4']], }, 'i m m m' => { number => 71, schoenflies => 'd_2h^25', full => 'i 2/m 2/m 2/m', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z']], }, 'i b a m' => { number => 72, schoenflies => 'd_2h^26', full => 'i 2/b 2/a 2/m', settings => ['i b a m', 'i m c b', 'i m c b', 'i c m a', 'i c m a'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z']], }, 'i b c a' => { number => 73, schoenflies => 'd_2h^27', full => '1 21/b 21/c 21/a', settings => ['i c a b', 'i b c a', 'i c a b', 'i b c a', 'i c a b'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z']], }, 'i m m a' => { number => 74, schoenflies => 'd_2h^28', full => 'i 21/m 21/m 21/a', settings => ['i m m b', 'i b m m', 'i c m m', 'i m c m', 'i m a m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z'], ['-$x', '$y+1/2', '-$z'], ['$x', '-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y+1/2', '-$z'], ['$x', '-$y+1/2', '$z'], ['-$x', '$y', '$z']], }, ## tetragonal groups 'p 4' => { number => 75, schoenflies => 'c_4^1', settings => ['c 4'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z']], }, 'p 41' => { number => 76, schoenflies => 'c_4^2', settings => ['c 41'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$y', '$x', '$z+1/4'], ['$y', '-$x', '$z+3/4']], }, 'p 42' => { number => 77, schoenflies => 'c_4^3', settings => ['c 42'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2']], }, 'p 43' => { number => 78, schoenflies => 'c_4^4', settings => ['c 43'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$y', '$x', '$z+3/4'], ['$y', '-$x', '$z+1/4']], }, 'i 4' => { number => 79, schoenflies => 'c_4^5', settings => ['f 4'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z']], }, 'i 41' => { number => 80, schoenflies => 'c_4^6', settings => ['f 41'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z+1/2'], ['-$y', '$x+1/2', '$z+1/4'], ['$y+1/2', '-$x', '$z+3/4']], }, 'p -4' => { number => 81, schoenflies => 's_4^1', settings => ['c -4'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z']], }, 'i -4' => { number => 82, schoenflies => 's_4^2', settings => ['f -4'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z']], }, 'p 4/m' => { number => 83, schoenflies => 'c_4h^1', settings => ['c 4/m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z']], }, 'p 42/m' => { number => 84, schoenflies => 'c_4h^2', settings => ['c 42/m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z+1/2'], ['-$y', '$x', '-$z+1/2']], }, 'p 4/n' => { number => 85, schoenflies => 'c_4h^3', settings => ['c 4/a'], shiftvec => [-1/4, 1/4, 0], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$y+1/2', '$x', '$z'], ['$y', '-$x+1/2', '$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$y+1/2', '-$x', '-$z'], ['-$y', '$x+1/2', '-$z']], }, 'p 42/n' => { number => 86, schoenflies => 'c_4h^4', settings => ['c 42/a'], shiftvec => [-1/4, -1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$y', '$x+1/2', '$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$y', '-$x+1/2', '-$z+1/2'], ['-$y+1/2', '$x', '-$z+1/2']], }, 'i 4/m' => { number => 87, schoenflies => 'c_4h^5', settings => ['f 4/m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z']], }, 'i 41/a' => { number => 88, schoenflies => 'c_4h^6', settings => ['f 41/d'], shiftvec => [0, -1/4, -1/8], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$y+3/4', '$x+1/4', '$z+1/4'], ['$y+3/4', '-$x+3/4', '$z+3/4'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$y+1/4', '-$x+3/4', '-$z+3/4'], ['-$y+1/4', '$x+1/4', '-$z+1/4']], }, 'p 4 2 2' => { number => 89, schoenflies => 'd_4^1', thirtyfive => 'p 4 2', settings => ['c 4 2 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z']], }, 'p 4 21 2' => { number => 90, schoenflies => 'd_4^2', thirtyfive => 'p 4 21', settings => ['c 4 21 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y+1/2', '$x+1/2', '$z'], ['$y+1/2', '-$x+1/2', '$z'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z']], }, 'p 41 2 2' => { number => 91, schoenflies => 'd_4^3', thirtyfive => 'p 41 2', settings => ['c 41 2 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$y', '$x', '$z+1/4'], ['$y', '-$x', '$z+3/4'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z+1/2'], ['$y', '$x', '-$z+3/4'], ['-$y', '-$x', '-$z+1/4']], }, 'p 41 21 2' => { number => 92, schoenflies => 'd_4^4', thirtyfive => 'p 41 21', settings => ['c 41 2 21'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$y+1/2', '$x+1/2', '$z+1/4'], ['$y+1/2', '-$x+1/2', '$z+3/4'], ['-$x+1/2', '$y+1/2', '-$z+1/4'], ['$x+1/2', '-$y+1/2', '-$z+3/4'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z+1/2']], }, 'p 42 2 2' => { number => 93, schoenflies => 'd_4^5', thirtyfive => 'p 42 2', settings => ['c 42 2 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$y', '$x', '-$z+1/2'], ['-$y', '-$x', '-$z+1/2']], }, 'p 42 21 2' => { number => 94, schoenflies => 'd_4^6', thirtyfive => 'p 42 21', settings => ['c 42 2 21'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y+1/2', '$x+1/2', '$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z+1/2'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z']], }, 'p 43 2 2' => { number => 95, schoenflies => 'd_4^7', thirtyfive => 'p 43 2', settings => ['c 43 2 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$y', '$x', '$z+3/4'], ['$y', '-$x', '$z+1/4'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z+1/2'], ['$y', '$x', '-$z+1/4'], ['-$y', '-$x', '-$z+3/4']], }, 'p 43 21 2' => { number => 96, schoenflies => 'd_4^8', thirtyfive => 'p 43 21', settings => ['c 43 2 21'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z+1/2'], ['-$y+1/2', '$x+1/2', '$z+3/4'], ['$y+1/2', '-$x+1/2', '$z+1/4'], ['-$x+1/2', '$y+1/2', '-$z+3/4'], ['$x+1/2', '-$y+1/2', '-$z+1/4'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z+1/2']], }, 'i 4 2 2' => { number => 97, schoenflies => 'd_4^9', thirtyfive => 'i 4 2', settings => ['f 4 2 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z']], }, 'i 41 2 2' => { number => 98, schoenflies => 'd_4^10', thirtyfive => 'i 41 2', settings => ['f 41 2 2'], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z+1/2'], ['-$y', '$x+1/2', '$z+1/4'], ['$y+1/2', '-$x', '$z+3/4'], ['-$x+1/2', '$y', '-$z+3/4'], ['$x', '-$y+1/2', '-$z+1/4'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y', '-$x', '-$z']], }, 'p 4 m m' => { number => 99, schoenflies => 'c_4v^1', settings => ['c 4 m m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'p 4 b m' => { number => 100, schoenflies => 'c_4v^2', settings => ['c 4 m g1'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z'], ['-$y+1/2', '-$x+1/2', '$z'], ['$y+1/2', '$x+1/2', '$z']], }, 'p 42 c m' => { number => 101, schoenflies => 'd_4v^3', settings => ['c 42 m c'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'p 42 n m' => { number => 102, schoenflies => 'c_4v^4', settings => ['c 42 m g2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y+1/2', '$x+1/2', '$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z+1/2'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'p 4 c c' => { number => 103, schoenflies => 'c_4v^5', settings => ['c 4 c c'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['$y', '$x', '$z+1/2']], }, 'p 4 n c' => { number => 104, schoenflies => 'p_4v^6', settings => ['c 4 c g2'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z+1/2'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'p 42 m c' => { number => 105, schoenflies => 'c_4v^7', settings => ['c 42 c m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$y', '-$x', '$z+1/2'], ['$y', '$x', '$z+1/2']], }, 'p 42 b c' => { number => 106, schoenflies => 'c_4v^8', settings => ['c 42 c g1'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'i 4 m m' => { number => 107, schoenflies => 'c_4v^9', settings => ['f 4 m m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'i 4 c m' => { number => 108, schoenflies => 'c_4v^10', settings => ['f 4 m c'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['$y', '$x', '$z+1/2']], }, 'i 41 m d' => { number => 109, schoenflies => 'c_4v^11', settings => ['f 41 d m'], # is that right? positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z+1/2'], ['-$y', '$x+1/2', '$z+1/4'], ['$y+1/2', '-$x', '$z+3/4'], ['$x', '-$y', '$z'], ['-$x+1/2', '$y+1/2', '$z+1/2'], ['-$y', '-$x+1/2', '$z+1/4'], ['$y+1/2', '$x', '$z+3/4']], }, 'i 41 c d' => { number => 110, schoenflies => 'c_4v^12', settings => ['f 41 d c'], # is that right? positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z+1/2'], ['-$y', '$x+1/2', '$z+1/4'], ['$y+1/2', '-$x', '$z+3/4'], ['$x', '-$y', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z'], ['-$y', '-$x+1/2', '$z+3/4'], ['$y+1/2', '$x', '$z+1/4']], }, 'p -4 2 m' => { number => 111, schoenflies => 'd_2d^1', settings => ['c -4 m 2'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'p -4 2 c' => { number => 112, schoenflies => 'd_2d^2', settings => ['c -4 c 2'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['$y', '$x', '$z+1/2']], }, 'p -4 21 m' => { number => 113, schoenflies => 'd_2d^3', settings => ['c -4 m 21'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z'], ['-$y+1/2', '-$x+1/2', '$z'], ['$y+1/2', '$x+1/2', '$z']], }, 'p -4 21 c' => { number => 114, schoenflies => 'd_2d^4', settings => ['c -4 c 21'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['-$x+1/2', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'p -4 m 2' => { number => 115, schoenflies => 'd_2d^5', settings => ['c -4 2 m'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z']], }, 'p -4 c 2' => { number => 116, schoenflies => 'd_2d^6', settings => ['c -4 2 c'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2'], ['$y', '$x', '-$z+1/2'], ['-$y', '-$x', '-$z+1/2']], }, 'p -4 b 2' => { number => 117, schoenflies => 'd_2d^7', settings => ['c -4 2 g1'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z'], ['$y+1/2', '$x+1/2', '-$z'], ['-$y+1/2', '-$x+1/2', '-$z']], }, 'p -4 n 2' => { number => 118, schoenflies => 'd_2d^8', settings => ['c -4 2 g2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2']], }, 'i -4 m 2' => { number => 119, schoenflies => 'd_2d^9', settings => ['f -4 2 m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z']], }, 'i -4 c 2' => { number => 120, schoenflies => 'd_2d^10', settings => ['f -4 2 c'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2'], ['$y', '$x', '-$z+1/2'], ['-$y', '-$x', '-$z+1/2']], }, 'i -4 2 m' => { number => 121, schoenflies => 'd_2d^11', settings => ['f -4 m 2'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'i -4 2 d' => { number => 122, schoenflies => 'd_2d^12', settings => ['f -4 d 2'], # is that right? positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['-$x+1/2', '$y', '-$z+3/4'], ['$x+1/2', '-$y', '-$z+3/4'], ['-$y+1/2', '-$x', '$z+3/4'], ['$y+1/2', '$x', '$z+3/4']], }, 'p 4/m m m' => { number => 123, schoenflies => 'd_4h^1', full => 'p 4/m 2/m 2/m', settings => ['c 4/m m m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'p 4/m c c' => { number => 124, schoenflies => 'd_4h^2', full => 'p 4/m 2/c 2/c', settings => ['c 4/m c c'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], # 2 ['-$y', '$x', '$z'], # 3 ['$y', '-$x', '$z'], # 4 ['-$x', '$y', '-$z+1/2'], # 5 ['$x', '-$y', '-$z+1/2'], # 6 ['$y', '$x', '-$z+1/2'], # 7 ['-$y', '-$x', '-$z+1/2'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x', '$y', '-$z'], # 10 ['$y', '-$x', '-$z'], # 11 ['-$y', '$x', '-$z'], # 12 ['$x', '-$y', '$z+1/2'], # 13 ['-$x', '$y', '$z+1/2'], # 14 ['-$y', '-$x', '$z+1/2'], # 15 ['$y', '$x', '$z+1/2']], # 16 }, 'p 4/n b m' => { # is that right? number => 125, schoenflies => 'd_4h^3', full => 'p 4/n 2/b 2/m', settings => ['c 4/a m b'], # is that right? shiftvec => [-1/4, -1/4, 0], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], # 2 ['-$y+1/2', '$x', '$z'], # 3 ['$y', '-$x+1/2', '$z'], # 4 ['-$x+1/2', '$y', '-$z'], # 5 ['$x', '-$y+1/2', '-$z'], # 6 ['$y', '$x', '-$z'], # 7 ['-$y+1/2', '-$x+1/2', '-$z'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x+1/2', '$y+1/2', '-$z'], # 10 ['$y+1/2', '-$x', '-$z'], # 11 ['-$y', '$x+1/2', '-$z'], # 12 ['$x+1/2', '-$y', '$z'], # 13 ['-$x', '$y+1/2', '$z'], # 14 ['-$y', '-$x', '$z'], # 15 ['$y+1/2', '$x+1/2', '$z']], }, 'p 4/n n c' => { number => 126, schoenflies => 'd_4h^4', full => 'p 4/n 2/n 2/c', settings => ['c 4/a c n'], shiftvec => [-1/4, -1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], # 2 ['-$y+1/2', '$x', '$z'], # 3 ['$y', '-$x+1/2', '$z'], # 4 ['-$x+1/2', '$y', '-$z+1/2'], # 5 ['$x', '-$y+1/2', '-$z+1/2'], # 6 ['$y', '$x', '-$z+1/2'], # 7 ['-$y+1/2', '-$x+1/2', '-$z+1/2'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x+1/2', '$y+1/2', '-$z'], # 10 ['$y+1/2', '-$x', '-$z'], # 11 ['-$y', '$x+1/2', '-$z'], # 12 ['$x+1/2', '-$y', '$z+1/2'], # 13 ['-$x', '$y+1/2', '$z+1/2'], # 14 ['-$y', '-$x', '$z+1/2'], # 15 ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'p 4/m b m' => { number => 127, schoenflies => 'd_4h^5', full => 'p 4/m 21/b 2/m', settings => ['c 4/m m b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], # 2 ['-$y', '$x', '$z'], # 3 ['$y', '-$x', '$z'], # 4 ['-$x+1/2', '$y+1/2', '-$z'], # 5 ['$x+1/2', '-$y+1/2', '-$z'], # 6 ['$y+1/2', '$x+1/2', '-$z'], # 7 ['-$y+1/2', '-$x+1/2', '-$z'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x', '$y', '-$z'], # 10 ['$y', '-$x', '-$z'], # 11 ['-$y', '$x', '-$z'], # 12 ['$x+1/2', '-$y+1/2', '$z'], # 13 ['-$x+1/2', '$y+1/2', '$z'], # 14 ['-$y+1/2', '-$x+1/2', '$z'], # 15 ['$y+1/2', '$x+1/2', '$z']], }, 'p 4/m n c' => { number => 128, schoenflies => 'd_4h^6', full => 'p 4/m 21/n 2/c', settings => ['c 4/m c n'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], # 2 ['-$y', '$x', '$z'], # 3 ['$y', '-$x', '$z'], # 4 ['-$x+1/2', '$y+1/2', '-$z+1/2'], # 5 ['$x+1/2', '-$y+1/2', '-$z+1/2'], # 6 ['$y+1/2', '$x+1/2', '-$z+1/2'], # 7 ['-$y+1/2', '-$x+1/2', '-$z+1/2'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x', '$y', '-$z'], # 10 ['$y', '-$x', '-$z'], # 11 ['-$y', '$x', '-$z'], # 12 ['$x+1/2', '-$y+1/2', '$z+1/2'], # 13 ['-$x+1/2', '$y+1/2', '$z+1/2'], # 14 ['-$y+1/2', '-$x+1/2', '$z+1/2'], # 15 ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'p 4/n m m' => { number => 129, schoenflies => 'd_4h^7', full => 'p 4/n 21/m 2/m', settings => ['c 4/a m m'], shiftvec => [1/4, -1/4, 0], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], # 2 ['-$y+1/2', '$x', '$z'], # 3 ['$y', '-$x+1/2', '$z'], # 4 ['-$x', '$y+1/2', '-$z'], # 5 ['$x+1/2', '-$y', '-$z'], # 6 ['$y+1/2', '$x+1/2', '-$z'], # 7 ['-$y', '-$x', '-$z'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x+1/2', '$y+1/2', '-$z'], # 10 ['$y+1/2', '-$x', '-$z'], # 11 ['-$y', '$x+1/2', '-$z'], # 12 ['$x', '-$y+1/2', '$z'], # 13 ['-$x+1/2', '$y', '$z'], # 14 ['-$y+1/2', '-$x+1/2', '$z'], # 15 ['$y', '$x', '$z']], }, 'p 4/n c c' => { number => 130, schoenflies => 'd_4h^8', full => 'p 4/n 21/c 2/c', settings => ['c 4/a c c'], shiftvec => [-1/4, -1/4, 0], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], # 2 ['-$y+1/2', '$x', '$z'], # 3 ['$y', '-$x+1/2', '$z'], # 4 ['-$x', '$y+1/2', '-$z+1/2'], # 5 ['$x+1/2', '-$y', '-$z+1/2'], # 6 ['$y+1/2', '$x+1/2', '-$z+1/2'], # 7 ['-$y', '-$x', '-$z+1/2'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x+1/2', '$y+1/2', '-$z'], # 10 ['$y+1/2', '-$x', '-$z'], # 11 ['-$y', '$x+1/2', '-$z'], # 12 ['$x', '-$y+1/2', '$z+1/2'], # 13 ['-$x+1/2', '$y', '$z+1/2'], # 14 ['-$y+1/2', '-$x+1/2', '$z+1/2'], # 15 ['$y', '$x', '$z+1/2']], }, 'p 42/m m c' => { number => 131, schoenflies => 'd_4h^9', full => 'p 42/m 2/m 2/c', settings => ['c 42/m c m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], # 2 ['-$y', '$x', '$z+1/2'], # 3 ['$y', '-$x', '$z+1/2'], # 4 ['-$x', '$y', '-$z'], # 5 ['$x', '-$y', '-$z'], # 6 ['$y', '$x', '-$z+1/2'], # 7 ['-$y', '-$x', '-$z+1/2'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x', '$y', '-$z'], # 10 ['$y', '-$x', '-$z+1/2'], # 11 ['-$y', '$x', '-$z+1/2'], # 12 ['$x', '-$y', '$z'], # 13 ['-$x', '$y', '$z'], # 14 ['-$y', '-$x', '$z+1/2'], # 15 ['$y', '$x', '$z+1/2']], }, 'p 42/m c m' => { number => 132, schoenflies => 'd_4h^10', full => 'p 42/m 2/c 2/m', settings => ['c 42/m m c'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], # 2 ['-$y', '$x', '$z+1/2'], # 3 ['$y', '-$x', '$z+1/2'], # 4 ['-$x', '$y', '-$z+1/2'], # 5 ['$x', '-$y', '-$z+1/2'], # 6 ['$y', '$x', '-$z'], # 7 ['-$y', '-$x', '-$z'], # 8 ['-$x', '-$y', '-$z'], # 9 ['$x', '$y', '-$z'], # 10 ['$y', '-$x', '-$z+1/2'], # 11 ['-$y', '$x', '-$z+1/2'], # 12 ['$x', '-$y', '$z+1/2'], # 13 ['-$x', '$y', '$z+1/2'], # 14 ['-$y', '-$x', '$z'], # 15 ['$y', '$x', '$z']], }, 'p 42/n b c' => { number => 133, schoenflies => 'd_4h^11', full => 'p 42/n 2/b 2/c', settings => ['c 42/a c b'], shiftvec => [-1/4, 1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$y+1/2', '$x', '$z+1/2'], ['$y', '-$x+1/2', '$z+1/2'], ['-$x+1/2', '$y', '-$z'], ['$x', '-$y+1/2', '-$z'], ['$y', '$x', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$y+1/2', '-$x', '-$z+1/2'], ['-$y', '$x+1/2', '-$z+1/2'], ['$x+1/2', '-$y', '$z'], ['-$x', '$y+1/2', '$z'], ['-$y', '-$x', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'p 42/n n m' => { number => 134, schoenflies => 'd_4h^12', full => 'p 42/n 2/n 2/m', settings => ['c 42/a m n'], shiftvec => [-1/4, 1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$y+1/2', '$x', '$z+1/2'], ['$y', '-$x+1/2', '$z+1/2'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '-$z+1/2'], ['$y', '$x', '-$z'], ['-$y+1/2', '-$x+1/2', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$y+1/2', '-$x', '-$z+1/2'], ['-$y', '$x+1/2', '-$z+1/2'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2'], ['-$y', '-$x', '$z'], ['$y+1/2', '$x+1/2', '$z']], }, 'p 42/m b c' => { number => 135, schoenflies => 'd_4h^13', full => 'p 42/m 21/b 2/c', settings => ['c 42/m c b'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z+1/2'], ['$y', '-$x', '$z+1/2'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y+1/2', '-$z'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z+1/2'], ['-$y', '$x', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y+1/2', '$z'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2']], }, 'p 42/m n m' => { number => 136, schoenflies => 'd_4h^14', full => 'p 42/m 21/n 2/m', settings => ['c 42/m m n'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y+1/2', '$x+1/2', '$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z+1/2'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$y+1/2', '$x+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z+1/2'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'p 42/n m c' => { number => 137, schoenflies => 'd_4h^15', full => 'p 42/n 21/m 2/c', settings => ['c 42/a c m'], shiftvec => [-1/4, 1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$y+1/2', '$x', '$z+1/2'], ['$y', '-$x+1/2', '$z+1/2'], ['-$x', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '-$z'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y', '-$x', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$y+1/2', '-$x', '-$z+1/2'], ['-$y', '$x+1/2', '-$z+1/2'], ['$x', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '$z'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y', '$x', '$z+1/2']], }, 'p 42/n c m' => { number => 138, schoenflies => 'd_4h^16', full => 'p 42/n 21/c 2/m', settings => ['c 42/a m c'], shiftvec => [-1/4, 1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$y+1/2', '$x', '$z+1/2'], ['$y', '-$x+1/2', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y', '-$z+1/2'], ['$y+1/2', '$x+1/2', '-$z'], ['-$y', '-$x', '-$z'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$y+1/2', '-$x', '-$z+1/2'], ['-$y', '$x+1/2', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y', '$z+1/2'], ['-$y+1/2', '-$x+1/2', '$z'], ['$y', '$x', '$z']], }, 'i 4/m m m' => { number => 139, schoenflies => 'd_4h^17', full => 'i 4/m 2/m 2/m', settings => ['f 4/m m m'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z']], }, 'i 4/m c m' => { number => 140, schoenflies => 'd_4h^18', full => 'i 4/m 2/c 2/m', settings => ['f 4/m m c'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$y', '$x', '$z'], ['$y', '-$x', '$z'], ['-$x', '$y', '-$z+1/2'], ['$x', '-$y', '-$z+1/2'], ['$y', '$x', '-$z+1/2'], ['-$y', '-$x', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '-$y', '$z+1/2'], ['-$x', '$y', '$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['$y', '$x', '$z+1/2']], }, 'i 41/a m d' => { number => 141, schoenflies => 'd_4h^19', full => 'i 41/a 2/m 2/d', settings => ['f 41/d d m'], shiftvec => [0, 1/4, -1/8], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$y+1/4', '$x+3/4', '$z+1/4'], ['$y+1/4', '-$x+1/4', '$z+3/4'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y', '-$z'], ['$y+1/4', '$x+3/4', '-$z+1/4'], ['-$y+1/4', '-$x+1/4', '-$z+3/4'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$y+3/4', '-$x+1/4', '-$z+3/4'], ['-$y+3/4', '$x+3/4', '-$z+1/4'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y', '$z'], ['-$y+3/4', '-$x+1/4', '$z+3/4'], ['$y+3/4', '$x+3/4', '$z+1/4']], }, 'i 41/a c d' => { number => 142, schoenflies => 'd_4h^20', full => 'i 41/a 2/c 2/d', settings => ['f 41/d d c'], shiftvec => [0, 1/4, -1/8], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$y+1/4', '$x+3/4', '$z+1/4'], ['$y+1/4', '-$x+1/4', '$z+3/4'], ['-$x+1/2', '$y', '-$z'], ['$x', '-$y', '-$z+1/2'], ['$y+1/4', '$x+3/4', '-$z+3/4'], ['-$y+1/4', '-$x+1/4', '-$z+1/4'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$y+3/4', '-$x+1/4', '-$z+3/4'], ['-$y+3/4', '$x+3/4', '-$z+1/4'], ['$x+1/2', '-$y', '$z'], ['-$x', '$y', '$z+1/2'], ['-$y+3/4', '-$x+1/4', '$z+1/4'], ['$y+3/4', '$x+3/4', '$z+3/4']], }, ## trigonal groups 'p 3' => { number => 143, schoenflies => 'c_3^1', positions => [ ['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'] ], }, 'p 31' => { number => 144, schoenflies => 'c_3^2', positions => [ ['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'] ], }, 'p 32' => { number => 145, schoenflies => 'c_3^3', positions => [ ['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'] ], }, 'r 3' => { number => 146, schoenflies => 'c_3^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'] ], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x']], }, 'p -3' => { number => 147, schoenflies => 'c_3i^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z']], }, 'r -3' => { number => 148, schoenflies => 'c_3i^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z']], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x'], ['-$x', '-$y', '-$z'], ['-$z', '-$x', '-$y'], ['-$y', '-$z', '-$x']], }, 'p 3 1 2' => { number => 149, schoenflies => 'd_3^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z']], }, 'p 3 2 1' => { number => 150, schoenflies => 'd_3^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z']], }, 'p 31 1 2' => { number => 151, schoenflies => 'd_3^3', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'], ['-$y', '-$x', '-$z+2/3'], ['-$x+$y', '$y', '-$z+1/3'], ['$x', '$x-$y', '-$z']], }, 'p 31 2 1' => { number => 152, schoenflies => 'd_3^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z+2/3'], ['-$x', '-$x+$y', '-$z+1/3']], }, 'p 32 1 2' => { number => 153, schoenflies => 'd_3^5', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'], ['-$y', '-$x', '-$z+1/3'], ['-$x+$y', '$y', '-$z+2/3'], ['$x', '$x-$y', '-$z']], }, 'p 32 2 1' => { number => 154, schoenflies => 'd_3^6', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z+1/3'], ['-$x', '-$x+$y', '-$z+2/3']], }, 'r 3 2' => { number => 155, schoenflies => 'd_3^7', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z']], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x'], ['-$y', '-$x', '-$z'], ['-$x', '-$z', '-$y'], ['-$z', '-$y', '-$x']], }, 'p 3 m 1' => { number => 156, schoenflies => 'c_3v^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z']], }, 'p 3 1 m' => { number => 157, schoenflies => 'c_3v^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p 3 c 1' => { number => 158, schoenflies => 'c_3v^3', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2']], }, 'p 3 1 c' => { number => 159, schoenflies => 'c_3v^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, 'r 3 m' => { number => 160, schoenflies => 'c_3v^5', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z']], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x'], ['$y', '$x', '$z'], ['$x', '$z', '$y'], ['$z', '$y', '$x']], }, 'r 3 c' => { number => 161, schoenflies => 'c_3v^6', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2']], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['$z+1/2', '$y+1/2', '$x+1/2']], }, 'p -3 1 m' => { number => 162, schoenflies => 'd_3d^1', full => 'p -3 1 2/m', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p -3 1 c' => { number => 163, schoenflies => 'd_3d^2', full => 'p -3 1 2/c', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$y', '-$x', '-$z+1/2'], ['-$x+$y', '$y', '-$z+1/2'], ['$x', '$x-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, 'p -3 m 1' => { number => 164, schoenflies => 'd_3d^3', full => 'p -3 2/m 1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z']], }, 'p -3 c 1' => { number => 165, full => 'p -3 2/c 1', schoenflies => 'd_3d^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '-$z+1/2'], ['$x-$y', '-$y', '-$z+1/2'], ['-$x', '-$x+$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2']], }, 'r -3 m' => { number => 166, schoenflies => 'd_3d^5', full => 'r -3 2/m', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z']], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x'], ['-$y', '-$x', '-$z'], ['-$x', '-$z', '-$y'], ['-$z', '-$y', '-$x'], ['-$x', '-$y', '-$z'], ['-$z', '-$x', '-$y'], ['-$y', '-$z', '-$x'], ['$y', '$x', '$z'], ['$x', '$z', '$y'], ['$z', '$y', '$x']], }, 'r -3 c' => { number => 167, schoenflies => 'd_3d^6', full => 'r -3 2/c', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$y', '$x', '-$z+1/2'], ['$x-$y', '-$y', '-$z+1/2'], ['-$x', '-$x+$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2']], rhombohedral => [['$x', '$y', '$z'], ['$z', '$x', '$y'], ['$y', '$z', '$x'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$x+1/2', '-$z+1/2', '-$y+1/2'], ['-$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$x', '-$y', '-$z'], ['-$z', '-$x', '-$y'], ['-$y', '-$z', '-$x'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['$z+1/2', '$y+1/2', '$x+1/2']], }, ## hexagonal groups 'p 6' => { number => 168, schoenflies => 'c_6^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z']], }, 'p 61' => { number => 169, schoenflies => 'c_6^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+5/6'], ['$x-$y', '$x', '$z+1/6']], }, 'p 65' => { number => 170, schoenflies => 'c_6^3', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/6'], ['$x-$y', '$x', '$z+5/6']], }, 'p 62' => { number => 171, schoenflies => 'c_6^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z+2/3'], ['$x-$y', '$x', '$z+1/3']], }, 'p 64' => { number => 172, schoenflies => 'c_6^5', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z+1/3'], ['$x-$y', '$x', '$z+2/3']], }, 'p 63' => { number => 173, schoenflies => 'c_6^6', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2']], }, 'p -6' => { number => 174, schoenflies => 'c_3h^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$x', '$y', '-$z'], ['-$y', '$x-$y', '-$z'], ['-$x+$y', '-$x', '-$z']], }, 'p 6/m' => { number => 175, schoenflies => 'c_6h^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$x', '$y', '-$z'], ['-$y', '$x-$y', '-$z'], ['-$x+$y', '-$x', '-$z']], }, 'p 63/m' => { number => 176, schoenflies => 'c_6h^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$x', '$y', '-$z+1/2'], ['-$y', '$x-$y', '-$z+1/2'], ['-$x+$y', '-$x', '-$z+1/2']], }, 'p 6 2 2' => { number => 177, schoenflies => 'd_6^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z']], }, 'p 61 2 2' => { number => 178, schoenflies => 'd_6^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+5/6'], ['$x-$y', '$x', '$z+1/6'], ['$y', '$x', '-$z+1/3'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z+2/3'], ['-$y', '-$x', '-$z+5/6'], ['-$x+$y', '$y', '-$z+1/2'], ['$x', '$x-$y', '-$z+1/6']], }, 'p 65 2 2' => { number => 179, schoenflies => 'd_6^3', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/6'], ['$x-$y', '$x', '$z+5/6'], ['$y', '$x', '-$z+2/3'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z+1/3'], ['-$y', '-$x', '-$z+1/6'], ['-$x+$y', '$y', '-$z+1/2'], ['$x', '$x-$y', '-$z+5/6']], }, 'p 62 2 2' => { number => 180, schoenflies => 'd_6^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+2/3'], ['-$x+$y', '-$x', '$z+1/3'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z+2/3'], ['$x-$y', '$x', '$z+1/3'], ['$y', '$x', '-$z+2/3'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z+1/3'], ['-$y', '-$x', '-$z+2/3'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z+1/3']], }, 'p 64 2 2' => { number => 181, schoenflies => 'd_6^5', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z+1/3'], ['-$x+$y', '-$x', '$z+2/3'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z+1/3'], ['$x-$y', '$x', '$z+2/3'], ['$y', '$x', '-$z+1/3'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z+2/3'], ['-$y', '-$x', '-$z+1/3'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z+2/3']], }, 'p 63 2 2' => { number => 182, schoenflies => 'd_6^6', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['-$y', '-$x', '-$z+1/2'], ['-$x+$y', '$y', '-$z+1/2'], ['$x', '$x-$y', '-$z+1/2']], }, 'p 6 m m' => { number => 183, schoenflies => 'c_6v^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p 6 c c' => { number => 184, schoenflies => 'c_6v^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, 'p 63 c m' => { number => 185, schoenflies => 'c_6v^3', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p 63 m c' => { number => 186, schoenflies => 'c_6v^4', shorthand => ['graphite', 'gra'], positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, 'p -6 m 2' => { number => 187, schoenflies => 'd_3h^1', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$x', '$y', '-$z'], ['-$y', '$x-$y', '-$z'], ['-$x+$y', '-$x', '-$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z']], }, 'p -6 c 2' => { number => 188, schoenflies => 'd_3h^2', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$x', '$y', '-$z+1/2'], ['-$y', '$x-$y', '-$z+1/2'], ['-$x+$y', '-$x', '-$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z']], }, 'p -6 2 m' => { number => 189, schoenflies => 'd_3h^3', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$x', '$y', '-$z'], ['-$y', '$x-$y', '-$z'], ['-$x+$y', '-$x', '-$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p -6 2 c' => { number => 190, schoenflies => 'd_3h^4', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['$x', '$y', '-$z+1/2'], ['-$y', '$x-$y', '-$z+1/2'], ['-$x+$y', '-$x', '-$z+1/2'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, 'p 6/m m m' => { number => 191, schoenflies => 'd_6h^1', full => 'p 6/m 2/m 2/m', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$x', '$y', '-$z'], ['-$y', '$x-$y', '-$z'], ['-$x+$y', '-$x', '-$z'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p 6/m c c' => { number => 192, schoenflies => 'd_6h^2', full => 'p 6/m 2/c 2/c', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z'], ['$y', '-$x+$y', '$z'], ['$x-$y', '$x', '$z'], ['$y', '$x', '-$z+1/2'], ['$x-$y', '-$y', '-$z+1/2'], ['-$x', '-$x+$y', '-$z+1/2'], ['-$y', '-$x', '-$z+1/2'], ['-$x+$y', '$y', '-$z+1/2'], ['$x', '$x-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$x', '$y', '-$z'], ['-$y', '$x-$y', '-$z'], ['-$x+$y', '-$x', '-$z'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, 'p 63/m c m' => { number => 193, schoenflies => 'd_6h^3', full => 'p 63/m 2/c 2/m', positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2'], ['$y', '$x', '-$z+1/2'], ['$x-$y', '-$y', '-$z+1/2'], ['-$x', '-$x+$y', '-$z+1/2'], ['-$y', '-$x', '-$z'], ['-$x+$y', '$y', '-$z'], ['$x', '$x-$y', '-$z'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$x', '$y', '-$z+1/2'], ['-$y', '$x-$y', '-$z+1/2'], ['-$x+$y', '-$x', '-$z+1/2'], ['-$y', '-$x', '$z+1/2'], ['-$x+$y', '$y', '$z+1/2'], ['$x', '$x-$y', '$z+1/2'], ['$y', '$x', '$z'], ['$x-$y', '-$y', '$z'], ['-$x', '-$x+$y', '$z']], }, 'p 63/m m c' => { number => 194, schoenflies => 'd_6h^4', full => 'p 63/m 2/m 2/c', shorthand => ['hex', 'hcp'], positions => [['$x', '$y', '$z'], ['-$y', '$x-$y', '$z'], ['-$x+$y', '-$x', '$z'], ['-$x', '-$y', '$z+1/2'], ['$y', '-$x+$y', '$z+1/2'], ['$x-$y', '$x', '$z+1/2'], ['$y', '$x', '-$z'], ['$x-$y', '-$y', '-$z'], ['-$x', '-$x+$y', '-$z'], ['-$y', '-$x', '-$z+1/2'], ['-$x+$y', '$y', '-$z+1/2'], ['$x', '$x-$y', '-$z+1/2'], ['-$x', '-$y', '-$z'], ['$y', '-$x+$y', '-$z'], ['$x-$y', '$x', '-$z'], ['$x', '$y', '-$z+1/2'], ['-$y', '$x-$y', '-$z+1/2'], ['-$x+$y', '-$x', '-$z+1/2'], ['-$y', '-$x', '$z'], ['-$x+$y', '$y', '$z'], ['$x', '$x-$y', '$z'], ['$y', '$x', '$z+1/2'], ['$x-$y', '-$y', '$z+1/2'], ['-$x', '-$x+$y', '$z+1/2']], }, ## cubic groups 'p 2 3' => { number => 195, schoenflies => 't^1', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x']], }, 'f 2 3' => { number => 196, schoenflies => 't^2', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x']], }, 'i 2 3' => { number => 197, schoenflies => 't^3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x']], }, 'p 21 3' => { number => 198, schoenflies => 't^4', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2']], }, 'i 21 3' => { number => 199, schoenflies => 't^5', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2']], }, 'p m -3' => { number => 200, schoenflies => 't_h^1', full => 'p 2/m -3', thirtyfive => 'p m 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x']], }, 'p n -3' => { number => 201, schoenflies => 't_h^2', full => 'p 2/n -3', thirtyfive => 'p n 3', shiftvec => [ -1/4, -1/4, -1/4 ], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '-$z+1/2'], ['$z', '$x', '$y'], ['$z', '-$x+1/2', '-$y+1/2'], ['-$z+1/2', '-$x+1/2', '$y'], ['-$z+1/2', '$x', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y+1/2', '$z', '-$x+1/2'], ['$y', '-$z+1/2', '-$x+1/2'], ['-$y+1/2', '-$z+1/2', '$x'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2'], ['-$z', '-$x', '-$y'], ['-$z', '$x+1/2', '$y+1/2'], ['$z+1/2', '$x+1/2', '-$y'], ['$z+1/2', '-$x', '$y+1/2'], ['-$y', '-$z', '-$x'], ['$y+1/2', '-$z', '$x+1/2'], ['-$y', '$z+1/2', '$x+1/2'], ['$y+1/2', '$z+1/2', '-$x']], }, 'f m -3' => { number => 202, schoenflies => 't_h^3', full => 'f 2/m -3', thirtyfive => 'f m 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x']], }, 'f d -3' => { number => 203, schoenflies => 't_h^4', full => 'p 2/d -3', thirtyfive => 'f d 3', shiftvec => [ -1/8, -1/8, -1/8 ], positions => [['$x', '$y', '$z'], ['-$x+1/4', '-$y+1/4', '$z'], ['-$x+1/4', '$y', '-$z+1/4'], ['$x', '-$y+1/4', '-$z+1/4'], ['$z', '$x', '$y'], ['$z', '-$x+1/4', '-$y+1/4'], ['-$z+1/4', '-$x+1/4', '$y'], ['-$z+1/4', '$x', '-$y+1/4'], ['$y', '$z', '$x'], ['-$y+1/4', '$z', '-$x+1/4'], ['$y', '-$z+1/4', '-$x+1/4'], ['-$y+1/4', '-$z+1/4', '$x'], ['-$x', '-$y', '-$z'], ['$x+3/4', '$y+3/4', '-$z'], ['$x+3/4', '-$y', '$z+3/4'], ['-$x', '$y+3/4', '$z+3/4'], ['-$z', '-$x', '-$y'], ['-$z', '$x+3/4', '$y+3/4'], ['$z+3/4', '$x+3/4', '-$y'], ['$z+3/4', '-$x', '$y+3/4'], ['-$y', '-$z', '-$x'], ['$y+3/4', '-$z', '$x+3/4'], ['-$y', '$z+3/4', '$x+3/4'], ['$y+3/4', '$z+3/4', '-$x']], }, 'i m -3' => { number => 204, schoenflies => 't_h^5', full => 'i 2/m -3', thirtyfive => 'i m 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x']], }, 'p a -3' => { number => 205, schoenflies => 't_h^6', full => 'p 2/a -3', thirtyfive => 'p a 3', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z'], ['-$z', '-$x', '-$y'], ['-$z+1/2', '$x+1/2', '$y'], ['$z+1/2', '$x', '-$y+1/2'], ['$z', '-$x+1/2', '$y+1/2'], ['-$y', '-$z', '-$x'], ['$y', '-$z+1/2', '$x+1/2'], ['-$y+1/2', '$z+1/2', '$x'], ['$y+1/2', '$z', '-$x+1/2']], }, 'i a -3' => { number => 206, schoenflies => 't_h^7', full => 'i 2/a -3', thirtyfive => 'i a 3', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z'], ['-$z', '-$x', '-$y'], ['-$z+1/2', '$x+1/2', '$y'], ['$z+1/2', '$x', '-$y+1/2'], ['$z', '-$x+1/2', '$y+1/2'], ['-$y', '-$z', '-$x'], ['$y', '-$z+1/2', '$x+1/2'], ['-$y+1/2', '$z+1/2', '$x'], ['$y+1/2', '$z', '-$x+1/2']], }, 'p 4 3 2' => { number => 207, schoenflies => 'o^1', thirtyfive => 'p 4 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['$y', '-$x', '$z'], ['-$y', '$x', '$z'], ['$x', '$z', '-$y'], ['-$x', '$z', '$y'], ['-$x', '-$z', '-$y'], ['$x', '-$z', '$y'], ['$z', '$y', '-$x'], ['$z', '-$y', '$x'], ['-$z', '$y', '$x'], ['-$z', '-$y', '-$x']], }, 'p 42 3 2' => { number => 208, schoenflies => 'o^2', thirtyfive => 'p 42 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['-$y+1/2', '$x+1/2', '$z+1/2'], ['$x+1/2', '$z+1/2', '-$y+1/2'], ['-$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x+1/2', '-$z+1/2', '$y+1/2'], ['$z+1/2', '$y+1/2', '-$x+1/2'], ['$z+1/2', '-$y+1/2', '$x+1/2'], ['-$z+1/2', '$y+1/2', '$x+1/2'], ['-$z+1/2', '-$y+1/2', '-$x+1/2']], }, 'f 4 3 2' => { number => 209, schoenflies => 'o^3', thirtyfive => 'f 4 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['$y', '-$x', '$z'], ['-$y', '$x', '$z'], ['$x', '$z', '-$y'], ['-$x', '$z', '$y'], ['-$x', '-$z', '-$y'], ['$x', '-$z', '$y'], ['$z', '$y', '-$x'], ['$z', '-$y', '$x'], ['-$z', '$y', '$x'], ['-$z', '-$y', '-$x']], }, 'f 41 3 2' => { number => 210, schoenflies => 'o^4', thirtyfive => 'f 41 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '-$z+1/2'], ['$z', '$x', '$y'], ['$z+1/2', '-$x', '-$y+1/2'], ['-$z', '-$x+1/2', '$y+1/2'], ['-$z+1/2', '$x+1/2', '-$y'], ['$y', '$z', '$x'], ['-$y+1/2', '$z+1/2', '-$x'], ['$y+1/2', '-$z', '-$x+1/2'], ['-$y', '-$z+1/2', '$x+1/2'], ['$y+3/4', '$x+1/4', '-$z+3/4'], ['-$y+1/4', '-$x+1/4', '-$z+1/4'], ['$y+1/4', '-$x+3/4', '$z+3/4'], ['-$y+3/4', '$x+3/4', '$z+1/4'], ['$x+3/4', '$z+1/4', '-$y+3/4'], ['-$x+3/4', '$z+3/4', '$y+1/4'], ['-$x+1/4', '-$z+1/4', '-$y+1/4'], ['$x+1/4', '-$z+3/4', '$y+3/4'], ['$z+3/4', '$y+1/4', '-$x+3/4'], ['$z+1/4', '-$y+3/4', '$x+3/4'], ['-$z+3/4', '$y+3/4', '$x+1/4'], ['-$z+1/4', '-$y+1/4', '-$x+1/4']], }, 'i 4 3 2' => { number => 211, schoenflies => 'o^5', thirtyfive => 'i 4 3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['$y', '-$x', '$z'], ['-$y', '$x', '$z'], ['$x', '$z', '-$y'], ['-$x', '$z', '$y'], ['-$x', '-$z', '-$y'], ['$x', '-$z', '$y'], ['$z', '$y', '-$x'], ['$z', '-$y', '$x'], ['-$z', '$y', '$x'], ['-$z', '-$y', '-$x']], }, 'p 43 3 2' => { number => 212, schoenflies => 'o^6', thirtyfive => 'p 43 3', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['$y+1/4', '$x+3/4', '-$z+3/4'], ['-$y+1/4', '-$x+1/4', '-$z+1/4'], ['$y+3/4', '-$x+3/4', '$z+1/4'], ['-$y+3/4', '$x+1/4', '$z+3/4'], ['$x+1/4', '$z+3/4', '-$y+3/4'], ['-$x+3/4', '$z+1/4', '$y+3/4'], ['-$x+1/4', '-$z+1/4', '-$y+1/4'], ['$x+3/4', '-$z+3/4', '$y+1/4'], ['$z+1/4', '$y+3/4', '-$x+3/4'], ['$z+3/4', '-$y+3/4', '$x+1/4'], ['-$z+3/4', '$y+1/4', '$x+3/4'], ['-$z+1/4', '-$y+1/4', '-$x+1/4']], }, 'p 41 3 2' => { number => 213, schoenflies => 'o^7', thirtyfive => 'p 41 3', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['$y+3/4', '$x+1/4', '-$z+1/4'], ['-$y+1/4', '-$x+1/4', '-$z+1/4'], ['$y+1/4', '-$x+1/4', '$z+3/4'], ['-$y+1/4', '$x+3/4', '$z+1/4'], ['$x+3/4', '$z+1/4', '-$y+1/4'], ['-$x+1/4', '$z+3/4', '$y+1/4'], ['-$x+3/4', '-$z+3/4', '-$y+3/4'], ['$x+1/4', '-$z+1/4', '$y+3/4'], ['$z+3/4', '$y+1/4', '-$x+1/4'], ['$z+1/4', '-$y+1/4', '$x+3/4'], ['-$z+1/4', '$y+3/4', '$x+1/4'], ['-$z+3/4', '-$y+3/4', '-$x+3/4']], }, 'i 41 3 2' => { number => 214, schoenflies => 'o^8', thirtyfive => 'i 41 3', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['$y+3/4', '$x+1/4', '-$z+1/4'], ['-$y+1/4', '-$x+1/4', '-$z+1/4'], ['$y+1/4', '-$x+1/4', '$z+3/4'], ['-$y+1/4', '$x+3/4', '$z+1/4'], ['$x+3/4', '$z+1/4', '-$y+1/4'], ['-$x+1/4', '$z+3/4', '$y+1/4'], ['-$x+3/4', '-$z+3/4', '-$y+3/4'], ['$x+1/4', '-$z+1/4', '$y+3/4'], ['$z+3/4', '$y+1/4', '-$x+1/4'], ['$z+1/4', '-$y+1/4', '$x+3/4'], ['-$z+1/4', '$y+3/4', '$x+1/4'], ['-$z+3/4', '-$y+3/4', '-$x+3/4']], }, 'p -4 3 m' => { number => 215, schoenflies => 't_d^1', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '$z'], ['-$y', '-$x', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '$z', '$y'], ['-$x', '$z', '-$y'], ['-$x', '-$z', '$y'], ['$x', '-$z', '-$y'], ['$z', '$y', '$x'], ['$z', '-$y', '-$x'], ['-$z', '$y', '-$x'], ['-$z', '-$y', '$x']], }, 'f -4 3 m' => { number => 216, schoenflies => 't_d^2', shorthand => ['zincblende', 'zns'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '$z'], ['-$y', '-$x', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '$z', '$y'], ['-$x', '$z', '-$y'], ['-$x', '-$z', '$y'], ['$x', '-$z', '-$y'], ['$z', '$y', '$x'], ['$z', '-$y', '-$x'], ['-$z', '$y', '-$x'], ['-$z', '-$y', '$x']], }, 'i -4 3 m' => { number => 217, schoenflies => 't_d^3', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '$z'], ['-$y', '-$x', '$z'], ['$y', '-$x', '-$z'], ['-$y', '$x', '-$z'], ['$x', '$z', '$y'], ['-$x', '$z', '-$y'], ['-$x', '-$z', '$y'], ['$x', '-$z', '-$y'], ['$z', '$y', '$x'], ['$z', '-$y', '-$x'], ['-$z', '$y', '-$x'], ['-$z', '-$y', '$x']], }, 'p -4 3 n' => { number => 218, schoenflies => 't_d^4', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$y+1/2', '$x+1/2', '-$z+1/2'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '$z+1/2', '-$y+1/2'], ['-$x+1/2', '-$z+1/2', '$y+1/2'], ['$x+1/2', '-$z+1/2', '-$y+1/2'], ['$z+1/2', '$y+1/2', '$x+1/2'], ['$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$z+1/2', '$y+1/2', '-$x+1/2'], ['-$z+1/2', '-$y+1/2', '$x+1/2']], }, 'f -4 3 c' => { number => 219, schoenflies => 't_d^5', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$y+1/2', '$x+1/2', '-$z+1/2'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '$z+1/2', '-$y+1/2'], ['-$x+1/2', '-$z+1/2', '$y+1/2'], ['$x+1/2', '-$z+1/2', '-$y+1/2'], ['$z+1/2', '$y+1/2', '$x+1/2'], ['$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$z+1/2', '$y+1/2', '-$x+1/2'], ['-$z+1/2', '-$y+1/2', '$x+1/2']], }, 'i -4 3 d' => { number => 220, schoenflies => 't_d^6', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['$y+1/4', '$x+1/4', '$z+1/4'], ['-$y+1/4', '-$x+3/4', '$z+3/4'], ['$y+3/4', '-$x+1/4', '-$z+3/4'], ['-$y+3/4', '$x+3/4', '-$z+1/4'], ['$x+1/4', '$z+1/4', '$y+1/4'], ['-$x+3/4', '$z+3/4', '-$y+1/4'], ['-$x+1/4', '-$z+3/4', '$y+3/4'], ['$x+3/4', '-$z+1/4', '-$y+3/4'], ['$z+1/4', '$y+1/4', '$x+1/4'], ['$z+3/4', '-$y+1/4', '-$x+3/4'], ['-$z+3/4', '$y+3/4', '-$x+1/4'], ['-$z+1/4', '-$y+3/4', '$x+3/4']], }, 'p m -3 m' => { number => 221, schoenflies => 'o_h^1', full => 'p 4/m -3 2/m', thirtyfive => 'p m 3 m', shorthand => ['cubic', 'cscl', 'perov', 'perovskite'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['$y', '-$x', '$z'], ['-$y', '$x', '$z'], ['$x', '$z', '-$y'], ['-$x', '$z', '$y'], ['-$x', '-$z', '-$y'], ['$x', '-$z', '$y'], ['$z', '$y', '-$x'], ['$z', '-$y', '$x'], ['-$z', '$y', '$x'], ['-$z', '-$y', '-$x'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z'], ['-$y', '$x', '-$z'], ['$y', '-$x', '-$z'], ['-$x', '-$z', '$y'], ['$x', '-$z', '-$y'], ['$x', '$z', '$y'], ['-$x', '$z', '-$y'], ['-$z', '-$y', '$x'], ['-$z', '$y', '-$x'], ['$z', '-$y', '-$x'], ['$z', '$y', '$x']], }, 'p n -3 n' => { number => 222, schoenflies => 'o_h^2', full => 'p 4/n -3 2/n', thirtyfive => 'p n 3 n', shiftvec => [-1/4, -1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '-$z+1/2'], ['$z', '$x', '$y'], ['$z', '-$x+1/2', '-$y+1/2'], ['-$z+1/2', '-$x+1/2', '$y'], ['-$z+1/2', '$x', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y+1/2', '$z', '-$x+1/2'], ['$y', '-$z+1/2', '-$x+1/2'], ['-$y+1/2', '-$z+1/2', '$x'], ['$y', '$x', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['$y', '-$x+1/2', '$z'], ['-$y+1/2', '$x', '$z'], ['$x', '$z', '-$y+1/2'], ['-$x+1/2', '$z', '$y'], ['-$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x', '-$z+1/2', '$y'], ['$z', '$y', '-$x+1/2'], ['$z', '-$y+1/2', '$x'], ['-$z+1/2', '$y', '$x'], ['-$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2'], ['-$z', '-$x', '-$y'], ['-$z', '$x+1/2', '$y+1/2'], ['$z+1/2', '$x+1/2', '-$y'], ['$z+1/2', '-$x', '$y+1/2'], ['-$y', '-$z', '-$x'], ['$y+1/2', '-$z', '$x+1/2'], ['-$y', '$z+1/2', '$x+1/2'], ['$y+1/2', '$z+1/2', '-$x'], ['-$y', '-$x', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['-$y', '$x+1/2', '-$z'], ['$y+1/2', '-$x', '-$z'], ['-$x', '-$z', '$y+1/2'], ['$x+1/2', '-$z', '-$y'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['-$x', '$z+1/2', '-$y'], ['-$z', '-$y', '$x+1/2'], ['-$z', '$y+1/2', '-$x'], ['$z+1/2', '-$y', '-$x'], ['$z+1/2', '$y+1/2', '$x+1/2']], }, 'p m -3 n' => { number => 223, schoenflies => 'o_h^3', thirtyfive => 'p m 3 n', full => 'p 42/m -3 2/n', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['-$y+1/2', '$x+1/2', '$z+1/2'], ['$x+1/2', '$z+1/2', '-$y+1/2'], ['-$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x+1/2', '-$z+1/2', '$y+1/2'], ['$z+1/2', '$y+1/2', '-$x+1/2'], ['$z+1/2', '-$y+1/2', '$x+1/2'], ['-$z+1/2', '$y+1/2', '$x+1/2'], ['-$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['-$y+1/2', '$x+1/2', '-$z+1/2'], ['$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$x+1/2', '-$z+1/2', '$y+1/2'], ['$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '$z+1/2', '-$y+1/2'], ['-$z+1/2', '-$y+1/2', '$x+1/2'], ['-$z+1/2', '$y+1/2', '-$x+1/2'], ['$z+1/2', '-$y+1/2', '-$x+1/2'], ['$z+1/2', '$y+1/2', '$x+1/2']], }, 'p n -3 m' => { number => 224, schoenflies => 'o_h^4', full => 'p 42/n -3 2/m', thirtyfive => 'p n 3 m', shiftvec => [-1/4, -1/4, -1/4], positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y+1/2', '$z'], ['-$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '-$z+1/2'], ['$z', '$x', '$y'], ['$z', '-$x+1/2', '-$y+1/2'], ['-$z+1/2', '-$x+1/2', '$y'], ['-$z+1/2', '$x', '-$y+1/2'], ['$y+1/2', '$z', '$x+1/2'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z+1/2', '$x'], ['$y+1/2', '$x+1/2', '-$z'], ['-$y', '-$x', '-$z'], ['$y+1/2', '-$x', '$z+1/2'], ['-$y', '$x+1/2', '$z+1/2'], ['$x+1/2', '$z+1/2', '-$y'], ['-$x', '$z+1/2', '$y+1/2'], ['-$x', '-$z', '-$y'], ['$x+1/2', '-$z', '$y+1/2'], ['$z+1/2', '$y+1/2', '-$x'], ['$z+1/2', '-$y', '$x+1/2'], ['-$z', '$y+1/2', '$x+1/2'], ['-$z', '-$y', '-$x'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y+1/2', '-$z'], ['$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '$z+1/2'], ['-$z', '-$x', '-$y'], ['-$z', '$x+1/2', '$y+1/2'], ['$z+1/2', '$x+1/2', '-$y'], ['$z+1/2', '-$x', '$y+1/2'], ['-$y', '-$z', '-$x'], ['$y+1/2', '-$z', '$x+1/2'], ['-$y', '$z+1/2', '$x+1/2'], ['$y+1/2', '$z+1/2', '-$x'], ['-$y+1/2', '-$x+1/2', '$z'], ['$y', '$x', '$z'], ['-$y+1/2', '$x', '-$z+1/2'], ['$y', '-$x+1/2', '-$z+1/2'], ['-$x+1/2', '-$z+1/2', '$y'], ['$x', '-$z+1/2', '-$y+1/2'], ['$x', '$z', '$y'], ['-$x+1/2', '$z', '-$y+1/2'], ['-$z+1/2', '-$y+1/2', '$x'], ['-$z+1/2', '$y', '-$x+1/2'], ['$z', '-$y+1/2', '-$x+1/2'], ['$z', '$y', '$x']], }, 'f m -3 m' => { number => 225, schoenflies => 'o_h^5', full => 'f 4/m -3 2/m', thirtyfive => 'f m 3 m', shorthand => ['fcc', 'salt', 'nacl'], positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['$y', '-$x', '$z'], ['-$y', '$x', '$z'], ['$x', '$z', '-$y'], ['-$x', '$z', '$y'], ['-$x', '-$z', '-$y'], ['$x', '-$z', '$y'], ['$z', '$y', '-$x'], ['$z', '-$y', '$x'], ['-$z', '$y', '$x'], ['-$z', '-$y', '-$x'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z'], ['-$y', '$x', '-$z'], ['$y', '-$x', '-$z'], ['-$x', '-$z', '$y'], ['$x', '-$z', '-$y'], ['$x', '$z', '$y'], ['-$x', '$z', '-$y'], ['-$z', '-$y', '$x'], ['-$z', '$y', '-$x'], ['$z', '-$y', '-$x'], ['$z', '$y', '$x']], }, 'f m -3 c' => { number => 226, schoenflies => 'o_h^6', full => 'f 4/m -3 2/c', thirtyfive => 'f m 3 c', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y+1/2', '$x+1/2', '-$z+1/2'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['$y+1/2', '-$x+1/2', '$z+1/2'], ['-$y+1/2', '$x+1/2', '$z+1/2'], ['$x+1/2', '$z+1/2', '-$y+1/2'], ['-$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x+1/2', '-$z+1/2', '$y+1/2'], ['$z+1/2', '$y+1/2', '-$x+1/2'], ['$z+1/2', '-$y+1/2', '$x+1/2'], ['-$z+1/2', '$y+1/2', '$x+1/2'], ['-$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x'], ['-$y+1/2', '-$x+1/2', '$z+1/2'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['-$y+1/2', '$x+1/2', '-$z+1/2'], ['$y+1/2', '-$x+1/2', '-$z+1/2'], ['-$x+1/2', '-$z+1/2', '$y+1/2'], ['$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+1/2', '$z+1/2', '-$y+1/2'], ['-$z+1/2', '-$y+1/2', '$x+1/2'], ['-$z+1/2', '$y+1/2', '-$x+1/2'], ['$z+1/2', '-$y+1/2', '-$x+1/2'], ['$z+1/2', '$y+1/2', '$x+1/2']], }, 'f d -3 m' => { number => 227, schoenflies => 'o_h^7', full => 'f 41/d -3 2/m', thirtyfive => 'f d 3 m', shorthand => ['diamond'], shiftvec => [-1/8, -1/8, -1/8], positions => [['$x', '$y', '$z'], ['-$x+3/4', '-$y+1/4', '$z+1/2'], ['-$x+1/4', '$y+1/2', '-$z+3/4'], ['$x+1/2', '-$y+3/4', '-$z+1/4'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+3/4', '-$y+1/4'], ['-$z+3/4', '-$x+1/4', '$y+1/2'], ['-$z+1/4', '$x+1/2', '-$y+3/4'], ['$y', '$z', '$x'], ['-$y+1/4', '$z+1/2', '-$x+3/4'], ['$y+1/2', '-$z+3/4', '-$x+1/4'], ['-$y+3/4', '-$z+1/4', '$x+1/2'], ['$y+3/4', '$x+1/4', '-$z+1/2'], ['-$y', '-$x', '-$z'], ['$y+1/4', '-$x+1/2', '$z+3/4'], ['-$y+1/2', '$x+3/4', '$z+1/4'], ['$x+3/4', '$z+1/4', '-$y+1/2'], ['-$x+1/2', '$z+3/4', '$y+1/4'], ['-$x', '-$z', '-$y'], ['$x+1/4', '-$z+1/2', '$y+3/4'], ['$z+3/4', '$y+1/4', '-$x+1/2'], ['$z+1/4', '-$y+1/2', '$x+3/4'], ['-$z+1/2', '$y+3/4', '$x+1/4'], ['-$z', '-$y', '-$x'], ['-$x', '-$y', '-$z'], ['$x+1/4', '$y+3/4', '-$z+1/2'], ['$x+3/4', '-$y+1/2', '$z+1/4'], ['-$x+1/2', '$y+1/4', '$z+3/4'], ['-$z', '-$x', '-$y'], ['-$z+1/2', '$x+1/4', '$y+3/4'], ['$z+1/4', '$x+3/4', '-$y+1/2'], ['$z+3/4', '-$x+1/2', '$y+1/4'], ['-$y', '-$z', '-$x'], ['$y+3/4', '-$z+1/2', '$x+1/4'], ['-$y+1/2', '$z+1/4', '$x+3/4'], ['$y+1/4', '$z+3/4', '-$x+1/2'], ['-$y+1/4', '-$x+3/4', '$z+1/2'], ['$y', '$x', '$z'], ['-$y+3/4', '$x+1/2', '-$z+1/4'], ['$y+1/2', '-$x+1/4', '-$z+3/4'], ['-$x+1/4', '-$z+3/4', '$y+1/2'], ['$x+1/2', '-$z+1/4', '-$y+3/4'], ['$x', '$z', '$y'], ['-$x+3/4', '$z+1/2', '-$y+1/4'], ['-$z+1/4', '-$y+3/4', '$x+1/2'], ['-$z+3/4', '$y+1/2', '-$x+1/4'], ['$z+1/2', '-$y+1/4', '-$x+3/4'], ['$z', '$y', '$x']], }, 'f d -3 c' => { number => 228, schoenflies => 'o_h^8', full => 'f 41/d -3 2/c', thirtyfive => 'f d 3 c', shiftvec => [ -3/8, -3/8, -3/8 ], positions => [['$x', '$y', '$z'], ['-$x+1/4', '-$y+3/4', '$z+1/2'], ['-$x+3/4', '$y+1/2', '-$z+1/4'], ['$x+1/2', '-$y+1/4', '-$z+3/4'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/4', '-$y+3/4'], ['-$z+1/4', '-$x+3/4', '$y+1/2'], ['-$z+3/4', '$x+1/2', '-$y+1/4'], ['$y', '$z', '$x'], ['-$y+3/4', '$z+1/2', '-$x+1/4'], ['$y+1/2', '-$z+1/4', '-$x+3/4'], ['-$y+1/4', '-$z+3/4', '$x+1/2'], ['$y+3/4', '$x+1/4', '-$z'], ['-$y+1/2', '-$x+1/2', '-$z+1/2'], ['$y+1/4', '-$x', '$z+3/4'], ['-$y', '$x+3/4', '$z+1/4'], ['$x+3/4', '$z+1/4', '-$y'], ['-$x', '$z+3/4', '$y+1/4'], ['-$x+1/2', '-$z+1/2', '-$y+1/2'], ['$x+1/4', '-$z', '$y+3/4'], ['$z+3/4', '$y+1/4', '-$x'], ['$z+1/4', '-$y', '$x+3/4'], ['-$z', '$y+3/4', '$x+1/4'], ['-$z+1/2', '-$y+1/2', '-$x+1/2'], ['-$x', '-$y', '-$z'], ['$x+3/4', '$y+1/4', '-$z+1/2'], ['$x+1/4', '-$y+1/2', '$z+3/4'], ['-$x+1/2', '$y+3/4', '$z+1/4'], ['-$z', '-$x', '-$y'], ['-$z+1/2', '$x+3/4', '$y+1/4'], ['$z+3/4', '$x+1/4', '-$y+1/2'], ['$z+1/4', '-$x+1/2', '$y+3/4'], ['-$y', '-$z', '-$x'], ['$y+1/4', '-$z+1/2', '$x+3/4'], ['-$y+1/2', '$z+3/4', '$x+1/4'], ['$y+3/4', '$z+1/4', '-$x+1/2'], ['-$y+1/4', '-$x+3/4', '$z'], ['$y+1/2', '$x+1/2', '$z+1/2'], ['-$y+3/4', '$x', '-$z+1/4'], ['$y', '-$x+1/4', '-$z+3/4'], ['-$x+1/4', '-$z+3/4', '$y'], ['$x', '-$z+1/4', '-$y+3/4'], ['$x+1/2', '$z+1/2', '$y+1/2'], ['-$x+3/4', '$z', '-$y+1/2'], ['-$z+1/4', '-$y+3/4', '$x'], ['-$z+3/4', '$y', '-$x+1/4'], ['$z', '-$y+1/4', '-$x+3/4'], ['$z+1/2', '$y+1/2', '$x+1/2']], }, 'i m -3 m' => { number => 229, schoenflies => 'o_h^9', full => 'i 4/m -3 2/m', shorthand => ['bcc'], thirtyfive => 'i m 3 m', positions => [['$x', '$y', '$z'], ['-$x', '-$y', '$z'], ['-$x', '$y', '-$z'], ['$x', '-$y', '-$z'], ['$z', '$x', '$y'], ['$z', '-$x', '-$y'], ['-$z', '-$x', '$y'], ['-$z', '$x', '-$y'], ['$y', '$z', '$x'], ['-$y', '$z', '-$x'], ['$y', '-$z', '-$x'], ['-$y', '-$z', '$x'], ['$y', '$x', '-$z'], ['-$y', '-$x', '-$z'], ['$y', '-$x', '$z'], ['-$y', '$x', '$z'], ['$x', '$z', '-$y'], ['-$x', '$z', '$y'], ['-$x', '-$z', '-$y'], ['$x', '-$z', '$y'], ['$z', '$y', '-$x'], ['$z', '-$y', '$x'], ['-$z', '$y', '$x'], ['-$z', '-$y', '-$x'], ['-$x', '-$y', '-$z'], ['$x', '$y', '-$z'], ['$x', '-$y', '$z'], ['-$x', '$y', '$z'], ['-$z', '-$x', '-$y'], ['-$z', '$x', '$y'], ['$z', '$x', '-$y'], ['$z', '-$x', '$y'], ['-$y', '-$z', '-$x'], ['$y', '-$z', '$x'], ['-$y', '$z', '$x'], ['$y', '$z', '-$x'], ['-$y', '-$x', '$z'], ['$y', '$x', '$z'], ['-$y', '$x', '-$z'], ['$y', '-$x', '-$z'], ['-$x', '-$z', '$y'], ['$x', '-$z', '-$y'], ['$x', '$z', '$y'], ['-$x', '$z', '-$y'], ['-$z', '-$y', '$x'], ['-$z', '$y', '-$x'], ['$z', '-$y', '-$x'], ['$z', '$y', '$x']], }, 'i a -3 d' => { number => 230, schoenflies => 'o_h^10', full => 'i 41/a -3 2/d', thirtyfive => 'i a 3 d', positions => [['$x', '$y', '$z'], ['-$x+1/2', '-$y', '$z+1/2'], ['-$x', '$y+1/2', '-$z+1/2'], ['$x+1/2', '-$y+1/2', '-$z'], ['$z', '$x', '$y'], ['$z+1/2', '-$x+1/2', '-$y'], ['-$z+1/2', '-$x', '$y+1/2'], ['-$z', '$x+1/2', '-$y+1/2'], ['$y', '$z', '$x'], ['-$y', '$z+1/2', '-$x+1/2'], ['$y+1/2', '-$z+1/2', '-$x'], ['-$y+1/2', '-$z', '$x+1/2'], ['$y+3/4', '$x+1/4', '-$z+1/4'], ['-$y+3/4', '-$x+3/4', '-$z+3/4'], ['$y+1/4', '-$x+1/4', '$z+3/4'], ['-$y+1/4', '$x+3/4', '$z+1/4'], ['$x+3/4', '$z+1/4', '-$y+1/4'], ['-$x+1/4', '$z+3/4', '$y+1/4'], ['-$x+3/4', '-$z+3/4', '-$y+3/4'], ['$x+1/4', '-$z+1/4', '$y+3/4'], ['$z+3/4', '$y+1/4', '-$x+1/4'], ['$z+1/4', '-$y+1/4', '$x+3/4'], ['-$z+1/4', '$y+3/4', '$x+1/4'], ['-$z+3/4', '-$y+3/4', '-$x+3/4'], ['-$x', '-$y', '-$z'], ['$x+1/2', '$y', '-$z+1/2'], ['$x', '-$y+1/2', '$z+1/2'], ['-$x+1/2', '$y+1/2', '$z'], ['-$z', '-$x', '-$y'], ['-$z+1/2', '$x+1/2', '$y'], ['$z+1/2', '$x', '-$y+1/2'], ['$z', '-$x+1/2', '$y+1/2'], ['-$y', '-$z', '-$x'], ['$y', '-$z+1/2', '$x+1/2'], ['-$y+1/2', '$z+1/2', '$x'], ['$y+1/2', '$z', '-$x+1/2'], ['-$y+1/4', '-$x+3/4', '$z+3/4'], ['$y+1/4', '$x+1/4', '$z+1/4'], ['-$y+3/4', '$x+3/4', '-$z+1/4'], ['$y+3/4', '-$x+1/4', '-$z+3/4'], ['-$x+1/4', '-$z+3/4', '$y+3/4'], ['$x+3/4', '-$z+1/4', '-$y+3/4'], ['$x+1/4', '$z+1/4', '$y+1/4'], ['-$x+3/4', '$z+3/4', '-$y+1/4'], ['-$z+1/4', '-$y+3/4', '$x+3/4'], ['-$z+3/4', '$y+3/4', '-$x+1/4'], ['$z+3/4', '-$y+1/4', '-$x+3/4'], ['$z+1/4', '$y+1/4', '$x+1/4']], }, ); my $thisdir = &identify_self; my $outfile = File::Spec -> catfile($thisdir, "space_groups.db"); (-e $outfile) && do { print " Removing old space_groups.db.\n"; unlink($outfile); }; print " Saving space_groups.db to disk.\n"; ##Storable: nstore(\%space_groups, $outfile) or die "can't store hash: $!\n"; ##end Storable: ##MLDBM: ## tie my %save_hash, 'MLDBM', "space_groups.db", O_CREAT|O_RDWR, 0644 or die $!; ## (tied %save_hash) -> DumpMeth('portable'); # Ask for portable binary ## %save_hash = %space_groups; ## untie %save_hash; ##end MLDBM: sub identify_self { my @caller = caller; use File::Basename qw(dirname); return dirname($caller[1]); }; 1; ## Local Variables: ## time-stamp-line-limit: 25 ## End: SpaceGroup/INSTALL0000644000175000017500000000117011017776331013160 0ustar segresegreQuick installation instructions for Xray::SpaceGroup Do these three steps at the command line: perl Build.PL ./Build sudo ./Build install If you are on a system that uses root, you should become root before doing the last step, then simply type ./Build install The first step will warn you if you have any missing dependencies. If you obtained this from the SVN repository, then you will find a script called "install_prereqs" in the directory above this one. Running it will install all the missing pre-requisites, including the Module::Build, which is used to build and install this module. BR (30 March, 2008)SpaceGroup/MANIFEST0000644000175000017500000000021311020062375013243 0ustar segresegreBuild.PL data/space_groups.db.PL example/sgreport.pl INSTALL lib/Xray/SpaceGroup.pm MANIFEST This list of files private-install t/test.t SpaceGroup/example/0000755000175000017500000000000011153052500013545 5ustar segresegreSpaceGroup/example/sgreport.pl0000755000175000017500000000122511030020047015745 0ustar segresegre#!/usr/bin/perl -I../blib/lib/ use Xray::SpaceGroup; use Term::ReadLine; my $term = new Term::ReadLine 'demeter'; my $sg = q{}; my $version = join("\n", q{}, "Xray::SpaceGroup version $Xray::SpaceGroup::VERSION,", "copyright (c) 2008 Bruce Ravel", "bravel AT bnl DOT gov", q{}); my $prompt = "Enter a space group symbol or q=quit > "; while ( defined ($_ = $term->readline($prompt)) ) { exit if ($_ =~ m{\Aq}i); next if ($_ =~ m{\A\s*\z}); print($version), next if ($_ =~ m{\Av}i); $sg = Xray::SpaceGroup -> new($_); if ($sg -> warning) { print "That wasn't a space group.\n"; } else { print $/, $sg -> report; }; }; SpaceGroup/Build.PL0000644000175000017500000000154211017776331013426 0ustar segresegreuse strict; use warnings; use Module::Build; my $build = Module::Build -> new( create_readme => 1, ##create_makefile_pl => 'traditional', license => 'perl', module_name => 'Xray::SpaceGroup', dist_author => 'Bruce Ravel ', dist_abstract => "Object oriented interface to a space groups database", requires => { 'version' => 0, 'Readonly' => 0, 'Class::Std' => 0, 'List::MoreUtils' => 0, 'Regexp::List' => 0, }, PL_files => { 'data/space_groups.db.PL' => 'data/space_groups.db', }, db_files => { 'data/space_groups.db' => 'lib/Xray/SpaceGroup/space_groups.db', }, recommends => {}, sign => 0, ); $build->add_build_element('db'); $build->create_build_script; SpaceGroup/MANIFEST.SKIP0000644000175000017500000000110311017776331014021 0ustar segresegre# Avoid version control files. \bRCS\b \bCVS\b ,v$ \B\.svn\b \B\.cvsignore$ # Avoid Makemaker generated and utility files. \bMakefile$ \bblib \bMakeMaker-\d \bpm_to_blib$ \bblibdirs$ ^MANIFEST\.SKIP$ # Avoid Module::Build generated and utility files. \bBuild$ \bBuild.bat$ \b_build # Avoid Devel::Cover generated files \bcover_db # Avoid temp and backup files. ~$ \.tmp$ \.old$ \.bak$ \#$ \.# \.rej$ # Avoid OS-specific files/dirs # Mac OSX metadata \B\.DS_Store # Mac OSX SMB mount metadata files \B\._ # Avoid archives of this distribution \bXray-SpaceGroup-[\d\.\_]+ SpaceGroup/private-install0000755000175000017500000000324411020062272015161 0ustar segresegre#!/bin/sh ###################################################################### ## This simple shell script is a replacement for the `make install' ## stage of the standard perl module installation sequence. If you ## are installing as a private user, you do not have write access to ## the system level directories where perl wants to install modules. ## It is possible to override these locations from the command line, ## but that requires a large number of awkward command line ## arguments. In this script, I have attempted to make good guesses ## for where things should be installed. These guesses are: ## ## modules: install to the location in the PERL5LIB or PERLLIB ## environment variable ## scripts: $HOME/bin -- a reasonable guess for where a normal user ## might keep personal programs and scripts ## man pages: into this directory -- this is roughly equivalent to ## not installing the man pages. most normal users do ## not keep personal man pages ## architecture dependent files: ## the auto/ directory under PERL5LIB or PERLLIB ## ## If this doesn't work for you, just edit this file as appropriate. ###################################################################### if [ $PERL5LIB ] then PLIB=$PERL5LIB elif [ $PERLLIB ] then PLIB=$PERLLIB else echo "Neither PERL5LIB nor PERLLIB are set. Private installation halting." exit fi ./Build --install_path lib=$PLIB \ --install_path arch=$PLIB \ --install_path bin=$HOME/bin \ --install_path script=$HOME/bin \ --install_path bindoc=`pwd`/man/ \ --install_path libdoc=`pwd`/man/ \ install