[GAP Forum] Steiner systems of Mathieu Group M12

[dmitrii.pasechnik at cs.ox.ac.uk]

Dear Forum,
On Tue, Jan 10, 2017 at 09:00:49PM +0000, PAUL HJELMSTAD wrote:
> Simply, is there a way to ask GAP to list all 5040 Steiner systems of the Mathieu Group M12?
yes, it is easy.
Get the hexads of one S(5,6,12), consider them as a "set of sets"
and compute the orbit of S_12 on them using OnSetsSets action.
In more detail:

gap> m12:=MathieuGroup(12);
gap> he:=Set(Orbit(m12,[1,2,3,4,5,7],OnSets));
gap> OrbitLength(SymmetricGroup(12),he,OnSetsSets);
5040


HTH,
Dmitrii
> 
> That is S12/[M12] Systems of hexads
> 
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