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In his paper 'Natural constructions of the Mathieu groups,' Curtis references an unpublished manuscript of G. Higman with a "significant constuction which makes use of the outer automorphism of $S_6$ to construct $M_{12},$ and the outer automorphism of $M_{12}$ to construct $M_{24}.$"

Does anyone know to what Curtis is referring? Is there another source for these constructions?

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I believe this is the construction in Bogopolsky's book:

You can read more about it if you look for the Higman-Sims group, etc.

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Unfortunately, Bogopolski only gives the definition of the Mathieu groups $M_{22},$ $M_{23},$ and $M_{24}$ so far as I can tell, and then only as automorphism groups of Steiner systems. Can you cite a page detailing the Higman construction? – Robert Haraway Jul 21 '11 at 3:19

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