Automorphism Groups of Steiner Systems
There exists up to equivalence a unique S(5,8,24) Steiner system W24 (the Witt design). The group M24 is the automorphism group of this Steiner system; that is, the set of permutations which map every block to some other block. The subgroups M23 and M22 are defined to be the stabilizers of a single point and two points respectively.
Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system W12, and the group M12 is its automorphism group. The subgroup M11 is the stabilizer of a poin
W12 can be constructed from the affine geometry on the vector space F3xF3, an S(2,3,9) system.
An alternative construction of W12 is the 'Kitten' of Curtis (1984).
An introduction to a construction of W24 via the Miracle Octad Generator of R. T. Curtis and Conway's analog for W12, the miniMOG, can be found in the book by Conway and Sloane.
Read more about this topic: Mathieu Group, Constructions of The Mathieu Groups
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