9 - The Mathieu Group M24

Summary

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 M_12, 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 M_24. 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 isomorphism

S6 : 2  ≅  PΓL2(9) ≅ A6.22

with 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.