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Published online by Cambridge University Press: 31 October 2024
This chapter is devoted to the smaller Mathieu group M12 that is the automorphism group of a Steiner system S(5, 6, 12). It possesses an outer automorphism group of order 2 and a group of shape M12 : 2 is a maximal subgroup of M12, the duum group described in Chapter 8. We introduce a device known as the Kitten, as it does for M12 what the MOG does for M24. Three copies of the 3 × 3 tic-tac-toe board are glued together to form a triangle in which the 132 hexads of the S(5, 6, 12) are readily recognized. The canonical embedding of M12 : 2 in M24 is described in detail. The symmetric group S6 is exceptional in that it possesses an outer automorphism; in this chapter we exhibit the isomorphismS6 : 2 ≅ PΓL2(9) ≅ A6.22with the group acting within M24 on (6+6)+2+10 letters. We digress from the general theme of this chapter to show how the beautiful Hoffman–Singleton graph that has 50 vertices and valency 7 appears neatly in the MOG.
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