Geometry of the Mathieu groups and Golay codes

Abstract

A brief review is given of the linear fractional subgroups of the Mathieu groups. The main part of the paper then deals with the projective interpretation of the Golay codes; these codes are shown to describe Coxeter's configuration in PG(5,3) and Todd’s configuration in PG(11,2) when interpreted projectively. We obtain two twelve-dimensional representations of$M_{24}$. One is obtained as the collineation group that permutes the twelve special points in PG(11,2); the other arises by interpreting geometrically the automorphism group of the binary Golay code. Both representations are reducible to eleven-dimensional representations of$M_{24}$.