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.
Categories: math.GR math.CO