[GAP Forum] extensions of subgroups of a finite 2-group

[Petr Savicky]

On Thu, Jul 17, 2014 at 08:54:41AM +0200, Benjamin Sambale wrote:
> Dear Petr,
> 
> I don't see how your first question is related to the group G. If you 
> want ALL extensions of A with a group of order 2, you could use 
> CyclicExtensions(A,2) from the GrpConst package. However, if A is small, 
> it is much faster to run through the groups of order 2|A| in the small 
> groups library and check which groups have maximal subgroups isomorphic 
> to A (i.e. the same GroupID).

Thank you for your reply. The extensions are considered as subgroups
of G and the embedding is important, not only the isomorphism type.
Consider the groups

  G := Group( [ (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16),
       (1,5)(2,6)(3,7)(4,8), (1,3)(2,4), (1,2) ] );

  A := Group( [ (3,4)(5,8,6,7)(11,12)(13,14), (3,4),
       (1,3)(2,4)(7,8)(11,12)(15,16), (3,4)(7,8)(9,12,10,11)(13,15,14,16),
       (13,14)(15,16) ] );

Group G has order 32768, group A has order 256 and is isomorphic
to SmallGroup(256, 27634).

There are 19 extensions B of A inside G with the quotient
group B/A = C_2. Using a precomputed list, they may be 
obtained as follows

  lst := [ (9,11,10,12)(13,15,14,16), (9,11)(10,12)(13,15)(14,16),
  (11,12)(15,16), (15,16), (11,12), (9,11)(10,12)(15,16),
  (11,12)(13,15)(14,16), (13,15)(14,16), (9,11)(10,12), (13,15,14,16),
  (9,11,10,12), (11,12)(13,15,14,16), (9,11,10,12)(15,16),
  (9,11)(10,12)(13,15,14,16), (9,11,10,12)(13,15)(14,16),
  (9,13)(10,14)(11,15)(12,16), (9,13)(10,14)(11,16)(12,15),
  (9,15)(10,16)(11,13)(12,14), (9,15,10,16)(11,14,12,13) ];
  B := [];
  for elm in lst do
      Add(B, ClosureGroup(A, elm));
  od;

Some of these extensions are conjugate. For example, the groups
B[16], ..., B[19] belong to the same conjugacy class.

Is it possible to find these groups using the small groups library?

I understand your suggestion as follows. Run through groups of
order 512 in the library, compute all MaximalSubgroups() or
ConjugacyClassesMaximalSubgroups() for each of them, then IdGroup()
for each of the maximal subgroups and compare with IdGroup(A).
Is this correct?

If we want to identify a maximal subgroup of SmallGroup(512, i)
in the library, is it possible to use somehow the fact that the
group was obtained as a maximal subgroup of a group from the library?

When the extensions of A are identified in the library, is it possible
to find all their embeddings into G?

Concerning the identification of the groups above, it is easy
to identify A using

  IdGroup(A);
  [ 256, 27634 ]

However, the groups B[i] have order 512 and I get an error
for them

  IdGroup(B[1]);
  Error, the group identification for groups of size 512 is not available called from
  <function "unknown">( <arguments> )

Is there a way, perhaps not a very efficient one, how to identify
the groups of order 512 in the library?

Best regards,
Petr.