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.



M12 simple sporadic group
You probably want
 [The golay codes and the Mathieu groups](http://books.google.com/books?id=upYwZ6cQumoC&pg=PA299) by John Conway


