The Mathieu group M-12 and its pseudogroup extension M-13

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.