The Mathieu Group $M_{12}$ and Its Pseudogroup Extension $M_{13}$

Abstract

We study a construction of the Mathieu group $M_{12}$ using a game reminiscent of Loyd's "15-puzzle.'' The elements of $M_{12}$ are realized as permutations on $12$ of the $13$ points of the finite projective plane of order $3$. There is a natural extension to a "pseudogroup'' $M_{13}$ acting on all $13$ points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both $M_{12}$ and $M_{13}$. We develop these results, and extend them to the double covers and automorphism groups of $M_{12}$ and $M_{13}$, using the ternary Golay code and $12 \x 12$ Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.