Conway, John H.; Elkies, Noam D.; Martin, Jeremy L. The Mathieu group \(M_{12}\) and its pseudogroup extension \(M_{13}\). (English) Zbl 1112.20003 Exp. Math. 15, No. 2, 223-236 (2006). Summary: 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 \(12\times 12\) 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. Cited in 1 ReviewCited in 7 Documents MSC: 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 51E20 Combinatorial structures in finite projective spaces 20D08 Simple groups: sporadic groups 94B25 Combinatorial codes 05B25 Combinatorial aspects of finite geometries 20B20 Multiply transitive finite groups Keywords:Mathieu groups; finite projective planes; Golay code; Hadamard matrices × Cite Format Result Cite Review PDF Full Text: DOI arXiv EuDML Link