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In his paper 'Natural constructions of the Mathieu groups,' Curtis references an unpublished manuscript of G. Higman with a "significant constuction which makes use of the outer automorphism of $S_6$ to construct $M_{12},$ and the outer automorphism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

http://books.google.com/books?id=jEw8MpP6DIgC&pg=PA39&lpg=PA39&dq=Higman+Mathieu+groups&source=bl&ots=g3vWN-s6ov&sig=frofirytVsXj-GLpMbqjo6sVShk&hl=en&ei=z2onTo-VBqHf0QHvttzfCg&sa=X&oi=book_result&ct=result&resnum=10&ved=0CFUQ6AEwCQ#v=onepage&q=Higman%20Mathieu%20groups&f=false

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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