[GAP Forum] Finding groups of a given order with a specific group as their Frattini Subgroup

[James Williams]

Hello,


I was wondering if somebody could advise me on the best method to achieve the following with GAP:


Given a group N and an order p^m, find all groups G of order p^m that have N as their Frattini subgroup.?


For example:

Let N be SmallGroup(3,1),  let p=3 and m=4.

Then I would like to find all groups G of order 81 with Frat(G)=N

In this case I expect to find SmallGroups [ 81, 11 ], [ 81, 12 ], [ 81, 13 ], [ 81, 14 ]


For small orders I am able to work backwards by filtering the small groups library, but I would like to be able to do this for large groups beyond the small groups library.?


I had wondered whether the Frattini Extension method in the GrpConst package would be useful here, but could not see how to use it to achieve this.


I would greatly appreciate any advice on this.


Thank you for your time,


James