[GAP Forum] Embeddings into a group

[dmitrii.pasechnik at cs.ox.ac.uk]

On Tue, May 01, 2018 at 01:45:13PM +0000, William Giuliano wrote:
>                      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?

Finding conjugacy classes of maximal subgroups ought to still be feasible,
and then you can do this sufficiently many times to get to order 2^{11}.

You might also try converting to a finitely presented group and doing
LowIndexSubgroupsFpGroup

HTH,
Dima

> 
> 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
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