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.