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I've spent quite a bit of time studying the Mathieu Groups, and I own the ATLAS.

My question is about M12. It is based on the ternary Golay Code, and is the automorphism group of a Steiner S(5,6,12) system. Now, all of these Steiner systems are isomorphic up to labelling. The order of M12 is 95040, which is 132 x 720. Since there are 132 blocks in this Steiner system, one can see that the 720 or S6 piece merely scrambles the six elements of the hexad. Then, the 132 part is just sending the elements of one hexad to another, of which there are 132 ways to do this.

Can someone give me an intuitive construction for this, not just generators...would it make sense to say that the (sharply) quintuple transitive action might be to send block 1 to 2, and 2 to 3, and perhaps another action to send block 1 to 3, 3 to 5 etc. or something of this nature? Is there a hard and fast way to look at this action (M12) in terms of the blocks? Or was I wrong about the 720 X 132 decomposition of the order of the group...Thanks, Paul.

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Apologies for the shameless plug, but I wrote a computer program related to M12 several years back (as well as M24 and Conway's .0 group) which may or may not be useful. The work was under the supervision of Prof. Igor Kriz, and the programs can be found on his homepage under "Sporadic Simple Puzzles": math.lsa.umich.edu/~ikriz. –  Paul Siegel Mar 23 '11 at 19:01
    
These are fun! Thanks. The 24 puzzle reminds me alot of the M13-game of John Conway, which also uses colored circles (26 of them). PGH –  Paul Hjelmstad Mar 23 '11 at 21:19
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up vote 7 down vote accepted

You probably want The golay codes and the Mathieu groups by John Conway

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Actually I have read that (and other chapters of SPLAG, which I own). hexacodeword construction is fun with 1,0,w,w-bar. I should probably be able to answer my own question with SPLAG, I just hoped perhaps there was a more direct method. PGH –  Paul Hjelmstad Mar 23 '11 at 20:25
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If you want an intuitive presentation of M12, also take a look at Curtis's construction.

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