[GAP Forum] Embeddings into a group

[Hulpke,Alexander]

Dear William, Dear Forum,

I fear the calculation you are attempting might be beyond the capabilities of `IsomorphicSubgroups`.

A few remarks on what is feasible:

> gap> iso:=IsomorphismPcGroup(B);;
> gap> emb1:=IsomorphicSubgroups(Image(iso),B1);;

At this point B1 is a group of automorphisms of order 2048 and Image(iso) a solvable group of order 16384.

I first would convert B1 into a pc group, as groups of automorphisms are somewhat unhandy.

Now B1 requires at least 6 generators, I am wary of >=4 for the algorithm used. Testing *will* take quite a while, regardless whether you want one or all.

I thus used a GAP 4.9 function, `LowLayerSubgroups(Image(iso),3)`

to determine subgroups of Image(iso) of order 2048. (Layer->Index works as we are in nilpotent groups).It finds about 6000 subgroups under conjugacy. Calculating orbit representatives under Aut(Image(iso)) reduces to ~250 and checking for properties such as Orders of upper central series subgroups, or orders of conjugacy classes reduced to 9 candidates.

Now, `RandomIsomorphismTest` established that all these groups are isomorphic to B1 (i.e there will be at least 9 different ways of embedding B1 into Image(iso)) , but does not give the actual isomorphism (but one could operate it out of the code with some effort). 

Do you acturally need the isomorphism or just the image subgroups?

Best wishes,

    Alexander Hulpke