R.T. Curtis in a 1990 paper* discussed his method of "symmetric generation" of groups as applied to the Mathieu groups
See Finite Relativity and the Log24 posts Relativity Problem Revisited (Sept. 20) and Symmetric Generation (Sept. 21).
Here is some exposition of how this works with
* "Geometric Interpretations of the ‘Natural’ Generators of the Mathieu groups," Mathematical Proceedings of the Cambridge Philosophical Society (1990), Vol. 107, Issue 01, pp. 19-26.