Subgroups in GAP

Question

GAP has the command ConjugacyClassesSubgroups which gives a list of the conjugacy classes of a finite group $G$. Is there a way I can specify further what types of subgroups GAP reports? For instance, can I ask GAP to only list conjugacy classes of subgroups of a certain order or isomorphism type?

My question is about subgroups in symmetric group isomorphic to others symmetric groups.

Thanks for your answers.

For instance, I have defined $S_3$ and $S_5$ and i would like to know the number of subgroups in $S_5$ isomorphic to $S_3$.

So the function IsomorphicSubgroups(s5,s3) enable to see 2 types of classes of subgroups in $S_5$ isomorphic to $S_3$. But how see the size of these classes?

If i put:

emb :=  IsomorphicSubgroups(s5,s3); 

And i ask:

Size(emb[1]);

GAP returns: "Error no method found!"

Thanks for any answer to help me!

Answer 1

IsomorphicSubgroups returns homomorphisms. The Image will be the subgroup that is isomorphic.

If you want the total number of subgroups that are isomorphic, the normalizer indices will give this

Sum(emb,x->Index(s5,Normalizer(s5,Image(x))));

However to understand the pattern, you might want to look at the image groups first.

Similarly for other groups replace S5, e.g.

gap> s6:=SymmetricGroup(6);;
gap> emb:=IsomorphicSubgroups(s6,s3);
[ [ (1,2,3), (1,2) ] -> [ (1,2,3), (1,2) ],
  [ (1,2,3), (1,2) ] -> [ (3,4,5), (1,2)(3,4) ],
  [ (1,2,3), (1,2) ] -> [ (1,2,5)(3,4,6), (1,2)(3,4) ],
  [ (1,2,3), (1,2) ] -> [ (1,3,5)(2,6,4), (1,2)(3,4)(5,6) ] ]
gap> Sum(emb,x->Index(s6,Normalizer(s6,Image(x))));
160