This is in regards to Chapter 11 of SPLAG. The tetracode construction of M12 is based
on col-col, col+tet, tet-tet, col+col-tet, which are 6 + 36 + 36 + 54 = 132. (Unsigned
hexads in the C12 code, of the Ternary Golay Code). Now I noticed the coincidence, that
the cols are the inverse of the tets in the C12 code, and also S3 in C4 X S3 is the inverse
of D4 in D4 X C3, in terms of generators, that is, S3 applied to a set will have the
inverse effect of D4 applied there. (And cols=3 elements while tets=4 elements). Perhaps
just a coincidence---


Anyway, 12P5 = 95040 and this is also the order of M12, the stabilizer of a S(5,6,12) Steiner
system, with the tetracode construction being one possible construction (with various labellings possible). Now I know that it is also a stabilizer of the C12 code, and that
M12 is quintuply sharply transitive on these sets. It sends blocks to blocks in S(5,6,12),
My question being, how a single g (member of) M12 accomplishes this, does each permutation
of 12P5, (which sends every pentad to every possible pentad) correspond to a g, so that
there are 95040 elements, acting regularly transitive on the Steiner system, and/or C12..


The best example would be the hexads (264) of the Ternary Golay Code, constructed with
the Tetracode, how does this stabilize C12?


Thanks 



M12 Simple Sporadic Group
I studied how M12 is built up from M9, in stages, and also got proficient with the tetracodeword construction. An interesting fact is that the Ternary Golay Code and the tetracodeword construction in SPLAG for S(5,6,12) (doubly = 264) are not so easily related as the Binary Golay Code and the hexacodeword construction for S(5,8,24), which is more direct.
This of course relates to the MiniMOG and the MOG, respectively. 


