[GAP Forum] Embeddings into a group

[William Giuliano]

Dear Forum,
                     I have constructed a group B of order 2^{14} (as an
automorphism group), in which I would like to try to embed other two groups
B1 and B2, both of order 2^{11}, in order to compute the number of double
cosets of B1 and B2 in B. I have tried to use “IsomorphicSubgroups” after
switching to a better representation of B (as a permutation and pc group),
but it didn’t work. I have read that for p-groups calculations can be slow,
but also finding the list of conjugacy classes of subgroups of B seems to
take too long. Does anyone have any suggestions?

gap> B:=AutomorphismGroup(G1234);
<group of size 16384 with 14 generators>
gap> F1:=AutomorphismGroup(G134);
<group of size 6144 with 10 generators>
gap> F2:=AutomorphismGroup(G234);
<group of size 6144 with 5 generators>
gap> A1:=[];;
gap> for f in F1 do
> if Image(f,G1234)=G1234 then Add(A1,f);fi;od;
gap> Size(A1);
2048
gap> A2:=[];;
gap> for f in F2 do
> if Image(f,G1234)=G1234 then Add(A2,f);fi;od;
gap> Size(A2);
2048
gap> B1:=Subgroup(F1,Elements(A1));;
gap> B2:=Subgroup(F2,Elements(A2));;
gap> Index(F1,B1);
3
gap> Index(F2,B2);
3
gap> iso:=IsomorphismPcGroup(B);;
gap> emb1:=IsomorphicSubgroups(Image(iso),B1);;
#I The group tested requires many generators. ‘IsomorphicSubgroups’ often
#I does not perform well for such groups -- see the documentation.

Thank you very much,
William