[GAP Forum] computing maximal transitive subgroups of a transitive group

[Stephen Glasby]

Dear Bill,

This is a good question. The answer is easy for n<4096. More generally,
what are good algorithms to compute
maximal primitive groups? or semiprimitive? or quasiprimitive? or rank 3
groups? ...

Given a transitive subgroup H of Sym(n), its overgroups are also
transitive. So you want
maximal subgroups of Sym(n) that are not conjugate to Sym(k)xSym(n-k).
These have been
classified. The imprimitive ones are Sym(k) wr Sym(n/k) and the primitive
ones are classified
by the O'Nan-Scott theorem. For the maximal (by inclusion)  affine
primitive groups of degree n=p^k you will need to know
maximal irreducible linear subgroups of GL(k,p). These are known for small
(k,p).
Thus you can get the complete answer for n<2^{12}=4096 as primitive groups
of this degree are classified (and in GAP and Magma).
Maybe this is good enough for you.

For background reading you may also want to look at the recent preprint
https://arxiv.org/pdf/1811.09015.pdf by Holt and Royle.
This builds on work of Hulpke, Cannon and others.

Best wishes,
Stephen Glasby


On Sun, 10 Nov 2019 at 07:17, Bill Allombert <
Bill.Allombert at math.u-bordeaux.fr> wrote:

> Dear GAP forum,
>
> I need to compute the maximal transitive subgroups of a transitive
> group (grouped by conjugacy classes).
>
> For example for S4, I should get A4 and two conjugate copies of D4.
>
> I managed to do the computation for all transitive groups of degree
> <=13, but for 14 it become quite slow.
>
> Are there good algorithms to solve this problem ?
>
> Thanks in advance for your help!
> Bill.
>
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