gap> uu:=(1,3)(2,4);ff:=(1,8)(4,5);dd:=(5,7)(6,8);
(1,3)(2,4)
(1,8)(4,5)
(5,7)(6,8)
gap> bb:=(3,6)(2,7);rr:=(2,5)(1,6);ll:=(4,7)(3,8);
(2,7)(3,6)
(1,6)(2,5)
(3,8)(4,7)
gap> H:=Group(uu,ff,rr,dd,bb,ll);
Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)
(4,7) )
gap>  H1:=Stabilizer(H,1);
Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), 
(3,8)(4,7) ), [ (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ] )
gap> Size(H1);
24
gap> GroupId(H1);
Error, the group library file 'groupid' must exist and be readable in
ReadGrp( "groupid" ) called from
main loop
brk> quit;
gap> N1:=NormalSubgroups(H1);
[ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)
    (3,6), (3,8)(4,7) ), [  ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), 
    (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (2,4)(5,7), (2,5)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), 
    (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (3,6,8)(4,5,7), (2,4,5)(3,8,6) ] ), Subgroup( Group( (1,3)(2,4), (1,8)
    (4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (5,7)(6,8), (2,5)(3,8), (2,4,5)(3,8,6) ] ) ]
gap> Length(N1);
4
gap> Size(N1[1]);
1
gap> Size(N1[2]);
4
gap> Size(N1[3]);
12
gap> Size(N1[4]);
24
gap> IsCyclic(N1[3]);
false
gap> IsDihedral(N1[3]);
Error, Variable: 'IsDihedral' must have a value
gap> IsAbelian(N1[3]);
false
gap> S4:=SymmetricGroup(4);
Error, the group library file 'basic' must exist and be readable in
ReadGrp( "basic" ) called from
main loop
brk> quit;
gap> S4:=Group((1,2),(1,2,3,4));
Group( (1,2), (1,2,3,4) )
gap> N2:=NormalSubgroups(S4);
[ Subgroup( Group( (1,2), (1,2,3,4) ), [  ] ), 
  Subgroup( Group( (1,2), (1,2,3,4) ), [ (1,2)(3,4), (1,4)(2,3) ] ), 
  Subgroup( Group( (1,2), (1,2,3,4) ), [ (2,3,4), (1,2,4) ] ), 
  Subgroup( Group( (1,2), (1,2,3,4) ), [ (3,4), (1,4), (2,4) ] ) ]
gap> Length(N2);
4
gap> Size(N2[1]);
1
gap> Size(N2[2]);
4
gap> Size(N2[3]);
12
gap> Size(N2[4]);
24
gap> IsAbelian(N2[3]);
false
gap> H3:=Group((1,2,3),(2,3,4));
Group( (1,2,3), (2,3,4) )
gap> Size(H3);
12
gap> (1,2,4)*(2,3,4);
(1,3,4)
gap> (2,3,4)*(1,2,4);
(1,2,3)
gap> N1[3];
Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), 
(3,8)(4,7) ), [ (3,6,8)(4,5,7), (2,4,5)(3,8,6) ] )
gap> Size(N1[3]);
12
gap> N3:=NormalSubgroups(N1[3]);
[ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)
    (3,6), (3,8)(4,7) ), [  ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), 
    (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (2,4)(5,7), (2,5)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), 
    (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (3,6,8)(4,5,7), (2,7,5)(3,6,8) ] ) ]
gap> Size(N3[1]);
1
gap> Size(N3[2]);
4
gap> Size(N3[3]);
12
gap> for g in N3[3] do Print(Order(N3[3],g)," "); od; Print("\n");
Error, for: <list> must evaluate to a list

gap> for g in GroupElements(N3[3]) do Print(Order(N3[3],g)," "); od; Print("\n");
Error, Function: <function> must be a function

gap> Order(N3[3],(3,6,8)(4,5,7));
3
gap> Order(N3[3],(3,6,8)(4,5,7)*(2,7,5)(3,6,8));
3
gap> Order(N3[3],(3,6,8)*(4,5,7)*(2,7,5)*(3,6,8));
3
gap> Order(N3[3],(3,6,8)^2*(4,5,7)^2*(2,7,5)*(3,6,8));
2
gap> Order(N3[3],(3,6,8)^2*(4,5,7)^2*(2,7,5)^2*(3,6,8)^2);
3
gap> Order(N3[3],(3,6,8)*(4,5,7)*(2,7,5)^2*(3,6,8)^2);
2
gap> N4:=NormalSubgroups(N3[3]);
[ Subgroup( Group( (1,3)(2,4), (1,8)(4,5), (1,6)(2,5), (5,7)(6,8), (2,7)
    (3,6), (3,8)(4,7) ), [  ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), 
    (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (2,4)(5,7), (2,5)(4,7) ] ), Subgroup( Group( (1,3)(2,4), (1,8)(4,5), 
    (1,6)(2,5), (5,7)(6,8), (2,7)(3,6), (3,8)(4,7) ), 
    [ (3,6,8)(4,5,7), (2,5,4)(3,6,8) ] ) ]
gap> Size(N4[1]);
1
gap> Size(N4[2]);
4
gap> Size(N4[3]);
12
gap> #This must be A4 since it has no elements of order 6
gap> #H must be S4 since it has a normal subgroup of index 2
gap> #isomorphic to A4 but is not a direct product
gap> IsElementaryAbelian(N4[2]);
true
gap> quit;