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gap> G := Group( [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ] ],
> [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
> [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ],
> [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] ] );
<matrix group with 4 generators>
gap> Size( G );
24
gap> BravaisGroup( G );
<matrix group of size 48 with 3 generators>
gap> L := BravaisSubgroups( G );
[ Group([ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ],
[ [ 0, 1, 0 ], [ 0, 0, -1 ], [ 1, 0, 0 ] ] ]),
Group([ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ]),
Group([ [ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ],
[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] ]),
<matrix group of size 16 with 3 generators>,
Group([ [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ],
[ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, 1 ] ] ]),
<matrix group of size 12 with 4 generators>,
<matrix group of size 8 with 3 generators>,
<matrix group of size 8 with 3 generators> ]
gap> List( L, IsBravaisGroup );
[ true, true, true, true, true, true, true, true ]
gap> BravaisSupergroups( G );
[ Group([ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ],
[ [ 0, 1, 0 ], [ 0, 0, -1 ], [ 1, 0, 0 ] ] ]) ]
gap> G = NormalizerInGLnZBravaisGroup( G );
true
gap> G = Normalizer( GL(3,Integers), G );
true
gap> IsTrivial( Centralizer(GL(3,Integers), G ) );
true
gap> L := ZClassRepsQClass( G );
[ <matrix group of size 24 with 4 generators>,
<matrix group of size 24 with 4 generators>,
<matrix group of size 24 with 4 generators> ]
gap> RepresentativeAction( GL(3,Integers), L[1], L[2] );
fail
gap> G2 := G^[[1,1,0],[1,0,0],[0,0,1]];
<matrix group of size 24 with 4 generators>
gap> C := RepresentativeAction( GL(3,Integers), G, G2 );
[ [ 0, 0, 1 ], [ 1, 1, 0 ], [ 1, 0, 0 ] ]
gap> G^C = G2;
true
gap> C := CaratQClassCatalog( G, 7 );
rec( familysymb := "3",
group := Group([ [ [ -1, 1, -1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ],
[ [ -1, 1, -1 ], [ 0, 0, -1 ], [ 1, 0, 0 ] ] ]),
qclass := "group.24",
trans := [ [ 1/2, 0, 1/2 ], [ -1/2, 1/2, 0 ], [ 0, -1/2, 1/2 ] ] )
gap> G^C.trans = C.group;
true
gap> C := ConjugatorQClass( L[1], L[3] );
[ [ -1/2, 0, -1/2 ], [ 0, 1/2, 0 ], [ -1/2, -1/2, 1/2 ] ]
gap> L[1]^C = L[3];
true
gap> G := Group( [ [[0,1,0,0],[0,0,1,0],[0,0,0,1],[-1,0,0,0]],
> [[-1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] ] );
<matrix group with 2 generators>
gap> L := BravaisSupergroups( G );
[ <matrix group of size 16 with 2 generators>,
<matrix group of size 384 with 3 generators>, <matrix group of size 1152 with
3 generators> ]
gap> LL := List( L, ZClassRepsQClass );
[ [ <matrix group of size 16 with 2 generators> ],
[ <matrix group of size 384 with 3 generators>,
<matrix group of size 384 with 3 generators>,
<matrix group of size 384 with 3 generators> ],
[ <matrix group of size 1152 with 3 generators> ] ]
gap> N := NormalizerInGLnZBravaisGroup( G );
<matrix group with 5 generators>
gap> GeneratorsOfGroup( N );
[ [ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ -1, 0, 0, 0 ] ],
[ [ -1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ] ],
[ [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ -1, 0, 0, 0 ] ],
[ [ 1, 1, 0, -1 ], [ -1, 0, 1, 1 ], [ 0, -1, -1, -1 ], [ 1, 1, 1, 0 ] ],
[ [ 0, 1, -1, 1 ], [ -1, 0, 1, -1 ], [ 1, -1, 0, 1 ], [ -1, 1, -1, 0 ] ] ]
gap> Size( N );
infinity
gap> C := ConjugatorQClass( LL[2][3], LL[2][2] );
[ [ 0, 1, 1, -1 ], [ -1/2, 1/2, 0, -1/2 ], [ 1/2, 0, -1/2, 1/2 ],
[ 0, -1/2, -1/2, 1 ] ]
gap> LL[2][3]^C = LL[2][2];
true
gap> CaratCrystalFamilies[5];
[ "1,1,1,1,1", "1,1,1,1;1", "1,1,1;1,1", "1,1,1;1;1", "1,1;1,1;1",
"1,1;1;1;1", "1;1;1;1;1", "2-1',2-1';1", "2-1,2-1;1",
"2-1;1,1,1", "2-1;1,1;1", "2-1;1;1;1", "2-1;2-1;1", "2-1;2-2;1",
"2-2',2-2';1", "2-2,2-2;1", "2-2;1,1,1", "2-2;1,1;1",
"2-2;1;1;1", "2-2;2-2;1", "3;1,1", "3;1;1", "3;2-1", "3;2-2",
"4-1';1", "4-1;1", "4-2';1", "4-2;1", "4-3';1", "4-3;1", "5-1",
"5-2" ]
gap> BravaisGroupsCrystalFamily( "4-2;1" );
[ <matrix group of size 576 with 4 generators>,
<matrix group of size 144 with 7 generators>,
<matrix group of size 288 with 4 generators>,
<matrix group of size 144 with 6 generators> ]
[ Dauer der Verarbeitung: 0.15 Sekunden
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