[GAP Forum] Capable extensions of 3-groups above SmallGroup(81,9)

[Siddhartha Sarkar]

Dear GAP forum,

My apologies to who is receiving this mail twice. I sent it to group pub
forum by mistake.

I am trying to cross check a few things about an extension of p-groups from
the available GAP library data (as there were different results in two
different sessions).

I am looking for 3-groups of coclass 1 starting from SmallGroup(81,9) that
possibly extend to a coclass 3 group modulo third central factor so that
Z3/Z2, Z2/Z both are C3xC3 and Z is C3 (Here C3 is cyclic group of order 3
and Zk is the kth upper centre, Z is centre).

Since these are capable, the Epicentre command gives these:

(1) (243,26) and (243,28) are the only ones lies above (81,9) by centre C3.
(2) (243,28) extends only by C9 (not by C3 or C3xC3)
(3) (729,95) only lies above (243,26) by C3 which can further extend
(4) There is only one group (2187,?) lies above (729,95) by C3 which can
further extend.

There were two different Epicentre outcomes for (729,96) which confused me.

This gives a sharp raise: (81,9)->(243,26)->(729,95)->(2187,?) of possible
members of G/Z3.

My apologies in advance that these might take several minutes.

Best regards,
Siddhartha