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?