The Mathieu group $M_{12}$ and its pseudogroup extension $M_{13}$

John H. Conway, Noam D. Elkies, Jeremy L. Martin

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…