Project by Shaochen, Daylene and Brian

gap> a:= [[1,2,3,4],[2,1,4,3], [3,4,1,2], [4,3,2,1]];
[ [ 1, 2, 3, 4 ], [ 2, 1, 4, 3 ], [ 3, 4, 1, 2 ], [ 4, 3, 2, 1 ] ]
gap> m:= MagmaByMultiplicationTable(a);

gap> g:= AsGroup(m);

gap> IsAbelian(g);
true
gap> IsCyclic(g);
false
gap> Display(MultiplicationTable(g)); 
[ [  1,  2,  3,  4 ],
  [  2,  1,  4,  3 ],
  [  3,  4,  1,  2 ],
  [  4,  3,  2,  1 ] ]
gap> 10  mod 5;
0
gap> 9 mod 5;
4
gap> Display(MultiplicationTable(m));
[ [  1,  2,  3,  4 ],
  [  2,  1,  4,  3 ],
  [  3,  4,  1,  2 ],
  [  4,  3,  2,  1 ] ]
gap> IsCyclic(m);
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `IsCyclic' on 1 arguments called from
(  ) 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> quit;
gap> l:=[(),(1,2,3)(4,5,6),(1,3,2)(4,6,5),(1,4)(2,5)(3,6),(1,5,3,4,2,6),(1,6,2,4,3,5)];
[ (), (1,2,3)(4,5,6), (1,3,2)(4,6,5), (1,4)(2,5)(3,6), (1,5,3,4,2,6), 
  (1,6,2,4,3,5) ]
gap> g:=AsGroup(l);
Group([ (1,2,3)(4,5,6), (1,4)(2,5)(3,6) ])
gap> Display(MultiplicationTable(g));
[ [  1,  2,  3,  4,  5,  6 ],
  [  2,  3,  1,  5,  6,  4 ],
  [  3,  1,  2,  6,  4,  5 ],
  [  4,  5,  6,  1,  2,  3 ],
  [  5,  6,  4,  2,  3,  1 ],
  [  6,  4,  5,  3,  1,  2 ] ]
gap> IsAbelian(g);
true
gap> Order(SymmetricGroup(8));
40320
gap> Order(DihedralGroup(8));
8
gap> s4:=SymmetricGroup(4);
Sym( [ 1 .. 4 ] )
gap> d8:=DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> d4:=DihedralGroup(IsPermGroup,8);
Group([ (1,2,3,4), (2,4) ])
gap> d8:=Subgroup(SymmetricGroup(8),[(1,2,3,4)(5,8,6,7),(1,5)(2,7)(3,6)(4,8)]);
Group([ (1,2,3,4)(5,8,6,7), (1,5)(2,7)(3,6)(4,8) ])
gap> d4:=Subgroup(SymmetricGroup(4),[(1,2,3,4),(1,2)(3,4)]);
Group([ (1,2,3,4), (1,2)(3,4) ])
gap> IsomorphismGroups(d4,d8);
[ (1,2,3,4), (1,2)(3,4) ] -> [ (1,2,3,4)(5,8,6,7), (1,8)(2,5)(3,7)(4,6) ]

Project by Dennis and Gordon S.

gap> d6:=DihedralGroup(IsPermGroup,12);
Group([ (1,2,3,4,5,6), (2,6)(3,5) ])
gap> c:=ConjugacyClasses(d6);
[ ()^G, (2,6)(3,5)^G, (1,2)(3,6)(4,5)^G, (1,2,3,4,5,6)^G, (1,3,5)(2,4,6)^G, 
  (1,4)(2,5)(3,6)^G ]
gap> Elements(c[1]);
[ () ]
gap> Elements(c[2]);
[ (2,6)(3,5), (1,3)(4,6), (1,5)(2,4) ]
gap> Elements(c[3]);
[ (1,2)(3,6)(4,5), (1,4)(2,3)(5,6), (1,6)(2,5)(3,4) ]
gap> Elements(c[4]);
[ (1,2,3,4,5,6), (1,6,5,4,3,2) ]
gap> Elements(c[5]);
[ (1,3,5)(2,4,6), (1,5,3)(2,6,4) ]
gap> Elements(c[6]);
[ (1,4)(2,5)(3,6) ]
gap> List(ConjugacyClasses(d6),Elements);
[ [ () ], [ (2,6)(3,5), (1,3)(4,6), (1,5)(2,4) ], 
  [ (1,2)(3,6)(4,5), (1,4)(2,3)(5,6), (1,6)(2,5)(3,4) ], 
  [ (1,2,3,4,5,6), (1,6,5,4,3,2) ], [ (1,3,5)(2,4,6), (1,5,3)(2,6,4) ], 
  [ (1,4)(2,5)(3,6) ] ]
gap> List(Size(ConjugacyClasses(d6),Elements));
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `Size' on 2 arguments called from
(  ) 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> quit;
gap> List(ConjugacyClasses(d6),Size);
[ 1, 3, 3, 2, 2, 1 ]
gap> Elements(Center(d6));
[ (), (1,4)(2,5)(3,6) ]
gap> Elements(Centre(d6));
[ (), (1,4)(2,5)(3,6) ]
gap> Order(Center(d4));
2
gap> s8:=SymmetricGroup(8);
Sym( [ 1 .. 8 ] )
gap> g:=SylowSubgroup(s8,2);
Group([ (1,2), (3,4), (1,3)(2,4), (5,6), (7,8), (5,7)(6,8), (1,5)(2,6)(3,7)(4,8) 
 ])
gap> Order(g);
128
gap> Order(Center(g));
2
gap> g:=SylowSubgroup(s8,3);
Group([ (1,2,3), (4,5,6) ])
gap> Order(g);
9
gap> Order(Center(g));
9
gap> g:=SylowSubgroup(s8,5);
Group([ (1,2,3,4,5) ])
gap> g:=SylowSubgroup(s8,7);
Group([ (1,2,3,4,5,6,7) ])
gap> s10:=SymmetricGroup(10);
Sym( [ 1 .. 10 ] )
gap> g:=SylowSubgroup(s10,2);
Group([ (1,2), (3,4), (1,3)(2,4), (5,6), (7,8), (5,7)(6,8), (1,5)(2,6)(3,7)(4,8),
  (9,10) ])
gap> Order(g);
256
gap> Order(Center(g));
4
gap> g:=SylowSubgroup(s10,3);
Group([ (1,2,3), (4,5,6), (7,8,9), (1,4,7)(2,5,8)(3,6,9) ])
gap> Order(g);
81
gap> Order(Center(g));
3
gap> g:=SylowSubgroup(s10,5);
Group([ (1,2,3,4,5), (6,7,8,9,10) ])
gap> Order(g);
25
gap> Order(Center(g));
25
gap> g:=SylowSubgroup(s10,7);
Group([ (1,2,3,4,5,6,7) ])
gap> Order(g);
7
gap> Order(Center(g));
7
gap> g2:=SylowSubgroup(s10,2);
Group([ (1,2), (3,4), (1,3)(2,4), (5,6), (7,8), (5,7)(6,8), (1,5)(2,6)(3,7)(4,8),
  (9,10) ])
gap> g22:=Subgroup(g2,[(1,2), (3,4)]);
Group([ (1,2), (3,4) ])
gap> Order(g2);
256
gap> Order(g22);
4