[GAP Forum] Listing Octads

[Alexander Hulpke]

Dear GAP forum,

Paul Hjelmstad wrote:

> Does anyone know how I would list:
Yes.

> 2) Representatives of Conjugacy Classes (26) in M24
gap> g:=MathieuGroup(24);
gap> List(ConjugacyClasses(g),Representative);
(Careful: Order is not necessarily ATLAS order but its easy to get a  
correspondence)

> 1) Octads in M24 (Are they Unique?)
The 759 octads are one orbit under M24 and are determined uniquely by  
5 points of this 5-transitive group, thus there are unique.

As the index of an octad stabilizer is 759 and thus odd, we can get  
one octad from orbits of a 2-Sylow subgroup:

gap> s:=SylowSubgroup(g,2);
<permutation group of size 1024 with 10 generators>
gap> Orbits(s,[1..24]);
[ [ 1, 5, 24, 8, 10, 11, 14, 20 ],
   [ 2, 13, 22, 19, 7, 3, 9, 23, 18, 4, 15, 12, 6, 16, 17, 21 ] ]
gap> o:=Set(last[1]);

The set of octads then is obtained as orbit:

gap> octads:=Orbit(g,o,OnSets);;
gap> Length(octads);
759

from which we see that in this labelling of M24 the lexicographically  
smallest octad is

gap> Set(octads)[1];
[ 1, 2, 3, 4, 5, 8, 11, 13 ]


Best,

    Alexander Hulpke


-- Colorado State University, Department of Mathematics,
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