Action of the Group M ₁₂ on Hadamard Matrices

Abstract

During recent years, interest in geometric characterizations of finite simple groups and, in particular, of sporadic groups has grown significantly [4], [5], [20]. In this respect a systematic investigation of the primitive representations of these groups is useful. In [18], [19] the lattices of 2-closed overgroups of the primitive representations of the nonabelian simple groups of order less then 10⁶ were studied with the help of computer calculations. These papers contain a number of interesting results presented in the form of computer output. So the problem to find a clear combinatorial-geometric interpretation of these results has arisen. As the simplest example of such an interpretation, a description of the 2-orbits of the primitive representation of the group M ₁₂ of degree 220 in terms of the Steiner system S (5,6,12) is obtained by I.V. Chuvaeva in [7]. In an analogous way, but with more involved constructions, it turns out to be possible to describe all primitive representations of the group M ₁₂, with the exception of the representation of degree 144 on the cosets of a subgroup isomorphic to L ₂ (11). Some fragments of this description can be found in [7]. The difference between the primitive representation of M ₁₂ on the cosets of L ₂(11) and all of its other primitive representations is mainly due to the fact that L ₂(11) is the only proper subgroup of M ₁₂, up to conjugacy, which acts primitively in both representations of M ₁₂ of degree 12. For this reason, the description in terms of the Steiner system S (5,6,12), which was very simple for other primitive representations, turns out to be too complicated in the case of the representation of degree 144.