[GAP Forum] Find a preserved partition given an imprimitive group action

[Nathann Cohen]

Helloooooo everybody !!!

I am but a humble graph theoretician, and I recently ended up playing with
the automorphism groups of my dear objects in order to find good ways to
plot them. This I do directly with GAP when I do not use it through Sage,
but I currently have a very technical problem and I would be delighted if
you happened to have already implemented somewhere the feature I am looking
for :-)

Here it is : I have a permutation group G, or a group acting on a set of
vertices in a graph, which sometime turns out to be imprimitive [1], which
means (I learned that recently) that there exists a nontrivial partition P
= P1, ... P_k of my set of vertices such that the  image of any P_i by an
element of G is another element of P. I do not know yet whether GAP can
tell me whether a given permutation group is primitive, but more than that
I would be interested in obtaining -- when it is not the case -- an example
of partition which is preserved in such a way.
Actually, because I am *very* greedy, I would ideally like to obtain an
inclusionwise "finest" (or smallest) partition, and in an ideal world to
enumerate them all.

May it be a task that GAP can solve ? :-)

Thank you very much for your help !

Nathann

[1] http://en.wikipedia.org/wiki/Primitive_permutation_group