PREP - Professional Enhancement Programs of the MAA
Abstract Algebra with GAP

A PREP Workshop

GAP log - Tuesday July 15, 2:30 pm - 4:00 pm

gap> S6:=SymmetricGroup(6);
Sym( [ 1 .. 6 ] )
gap> A6:=AlternatingGroup(6);
Alt( [ 1 .. 6 ] )
gap> D6:=DihedralGroup(IsPermGroup,12);
Group([ (1,2,3,4,5,6), (2,6)(3,5) ])
gap> Z6:=Center(D6);
Group([ (1,4)(2,5)(3,6) ])
gap> IsNormal(S6,A6);
true
gap> IsNormal(S6,D6);
false
gap> IsNormal(D6,Z6);
true
gap> RightCosets(S6,A6);
[ RightCoset(AlternatingGroup( [ 1 .. 6 ] ),()), RightCoset(AlternatingGroup( [ 1 .. 6 ] ),(5,6)) ]
gap> LeftCosets(S6,A6);
Variable: 'LeftCosets' must have a value

gap> Size(FactorGroup(S6,A6));
2
gap> RightCosets(D6,Z6);
[ RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),()), RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(2,6)(3,5)), 
  RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,5,3)(2,6,4)), RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,5)(2,4)), 
  RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,3,5)(2,4,6)), RightCoset(Group( [ ( 1, 4)( 2, 5)( 3, 6) ] ),(1,3)(4,6)) ]
gap> F:=FactorGroup(D6,Z6);
Group([ f1, f2 ])
gap> IsAbelian(F);
false
gap> Read("orderFrequency");
gap> orderFrequency(F);
[ [ 1, 1 ], [ 2, 3 ], [ 3, 2 ] ]
gap> orderFrequency(SymmetricGroup(3));
[ [ 1, 1 ], [ 2, 3 ], [ 3, 2 ] ]
gap> RightCosets(S6,D6);
[ RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),()), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(4,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(4,5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(4,6,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(4,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,4)(5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,4,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,4,5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,4,6,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,4,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,5,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,5,6,4)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,5)(4,6)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,5,4,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,6,5,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,6,4)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,6,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,6,4,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(3,6)(4,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3)(5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3)(4,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3)(4,5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3)(4,6,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3)(4,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,4)(5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,4,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,4,6,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,5,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,5,6,4)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,5)(4,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,6,5,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,6,4)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,6,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,3,6,4,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,3)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,3)(5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,5,3)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,6,5,3)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4)(5,6)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,6,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4)(3,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,3,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4)(3,6,5)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,4,3,6,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,5,4,3)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,5,3)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,5,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,5)), 
  RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,5,3,4)), RightCoset(Group( [ (1,2,3,4,5,6), (2,6)(3,5) ] ),(2,5)(3,4)) ]
gap> Index(S6,D6);
60
gap> Elements(RightCoset(Z6,(2,6)(3,6)));
Permutation: cycles must be disjoint and duplicate-free
gap> Elements(RightCoset(Z6,(2,6)(3,5)));
[ (2,6)(3,5), (1,4)(2,3)(5,6) ]
gap> D8 := DihedralGroup(IsPermGroup,16);
Group([ (1,2,3,4,5,6,7,8), (2,8)(3,7)(4,6) ])
gap> Z8 := Center(D8);
Group([ (1,5)(2,6)(3,7)(4,8) ])
gap> RightCosets(D8,Z8);
[ RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),()), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(2,8)(3,7)(4,6)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,7,5,3)(2,8,6,4)), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,7)
    (2,6)(3,5)), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,8,7,6,5,4,3,2)), 
  RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,8)(2,7)(3,6)(4,5)), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),
    (1,6,3,8,5,2,7,4)), RightCoset(Group( [ ( 1, 5)( 2, 6)( 3, 7)( 4, 8) ] ),(1,6)(2,5)(3,4)(7,8)) ]
gap> F:=FactorGroup(D8,Z8);
Group([ f1, f2, f3 ])
gap> e:=Elements(F);
[ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ]
gap> PrintArray(MultiplicationTable(e));
[ [  1,  2,  3,  4,  5,  6,  7,  8 ],
  [  2,  1,  5,  6,  3,  4,  8,  7 ],
  [  3,  8,  4,  7,  2,  5,  1,  6 ],
  [  4,  6,  7,  1,  8,  2,  3,  5 ],
  [  5,  7,  6,  8,  1,  3,  2,  4 ],
  [  6,  4,  8,  2,  7,  1,  5,  3 ],
  [  7,  5,  1,  3,  6,  8,  4,  2 ],
  [  8,  3,  2,  5,  4,  7,  6,  1 ] ]
gap> orderFrequency(F);
[ [ 1, 1 ], [ 2, 5 ], [ 4, 2 ] ]
gap> orderFrequency(DihedralGroup(8));
[ [ 1, 1 ], [ 2, 5 ], [ 4, 2 ] ]
gap> D10 := DihedralGroup(IsPermGroup,20);
Group([ (1,2,3,4,5,6,7,8,9,10), (2,10)(3,9)(4,8)(5,7) ])
gap> Z10 := Center(D10);
Group([ (1,6)(2,7)(3,8)(4,9)(5,10) ])
gap> F:=FactorGroup(D10,Z10);
Group([ f1, f2 ])
gap> Size(F);
10
gap> orderFrequency(DihedralGroup(10));
[ [ 1, 1 ], [ 2, 5 ], [ 5, 4 ] ]
gap> orderFrequency(F);
[ [ 1, 1 ], [ 2, 5 ], [ 5, 4 ] ]
gap> D12 := DihedralGroup(IsPermGroup,24);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12), (2,12)(3,11)(4,10)(5,9)(6,8) ])
gap> Z12 := Center(D12);
Group([ (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) ])
gap> F:=FactorGroup(D12,Z12);
Group([ f1, f2, f3 ])
gap> Size(F);
12
gap> orderFrequency(F);
[ [ 1, 1 ], [ 2, 7 ], [ 3, 2 ], [ 6, 2 ] ]
gap> orderFrequency(DihedralGroup(12));
[ [ 1, 1 ], [ 2, 7 ], [ 3, 2 ], [ 6, 2 ] ]
gap> S3:= SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> f1:= GroupHomomorphismByImages(S3,S3,[(1,2,3),(1,3)], [(1,3,2), (1,2)]);
[ (1,2,3), (1,3) ] -> [ (1,3,2), (1,2) ]
gap> Image(f1, (2,3));
(2,3)
gap> Image(f1, (1,2));
(1,3)
gap> Size(Image(f1));
6
gap> Kernel(f1);
Group(())
gap> f2:= GroupHomomorphismByImages(S3,S3,[(1,2,3),(1,3)], [(), (1,2)]);
[ (1,2,3), (1,3) ] -> [ (), (1,2) ]
gap> Kernel(f2);
Group([ (1,2,3) ])
gap> f3:= GroupHomomorphismByImages(S3,S3,[(1,2,3),(1,3)], [(2,3), (1,2)]);
fail
gap> Read("autoDn");
gap> Read("homoDn");
gap> d6:= DihedralGroup(IsPermGroup, 12);
Group([ (1,2,3,4,5,6), (2,6)(3,5) ])
gap> autoDn(d6);
[ [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,2)(3,6)(4,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,4)(2,3)(5,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,2)(3,6)(4,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,4)(2,3)(5,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,6)(2,5)(3,4) ] ]
gap> homoDn(d6);
[ [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), (1,4)(2,3)(5,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (2,6)(3,5), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2)(3,6)(4,5), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2)(3,6)(4,5), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,2)(3,6)(4,5), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2)(3,6)(4,5), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,2)(3,6)(4,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,4)(2,3)(5,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,2,3,4,5,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3)(4,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,2)(3,6)(4,5) ]
    , [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,3,5)(2,4,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,3,5)(2,4,6), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,3)(5,6), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,3)(5,6), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,3)(5,6), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,3)(5,6), (1,4)(2,5)(3,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (2,6)(3,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,5)(3,6), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,4)(2,3)(5,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,4)(2,5)(3,6), (1,5)(2,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,4)(2,5)(3,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), () ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), (1,2)(3,6)(4,5) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5)(2,4), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (2,6)(3,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,2)(3,6)(4,5) ]
    , [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,5,3)(2,6,4), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (2,6)(3,5) ],
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,2)(3,6)(4,5) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,3)(4,6) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,4)(2,3)(5,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,5)(2,4) ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6,5,4,3,2), (1,6)(2,5)(3,4) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6)(2,5)(3,4), () ], 
  [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6)(2,5)(3,4), (1,3)(4,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> 
    [ (1,6)(2,5)(3,4), (1,4)(2,5)(3,6) ], [ (1,2,3,4,5,6), (2,6)(3,5) ] -> [ (1,6)(2,5)(3,4), (1,6)(2,5)(3,4) ] ]
gap> d4:= DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> autoDn(d4);
[ [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,2,3,4), (1,4)(2,3) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (2,4) ], [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,2)(3,4) ], 
  [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,3) ], [ (1,2,3,4), (2,4) ] -> [ (1,4,3,2), (1,4)(2,3) ] ]
gap> Size(homoDn(d4));
36
gap> 
gap> Size(homoDn(DihedralGroup(IsPermGroup,10)));
26
gap> Size(homoDn(DihedralGroup(IsPermGroup,38)));
362
gap> Size(homoDn(DihedralGroup(IsPermGroup,42)));
442
gap> Size(autoDn(DihedralGroup(IsPermGroup,5)));
Error, <2n> must be an even integer called from
DihedralGroupCons( arg[1], arg[2] ) called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> gap> Size(autoDn(DihedralGroup(IsPermGroup,10)));
20
gap> Size(autoDn(DihedralGroup(IsPermGroup,38)));
342
gap> LogTo();
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This PREP workshop is made possible by the NSF grant DUE: 0089005