The double cover had been implicitly found earlier by Coxeter (1958), who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements.
The double cover 2.M12 is the automorphism group of the extended ternary Golay code, a dimension 6 length 12 code over the field of order 3 of minimum weight 6.
The double cover 2.M12 is the automorphism group of any 12×12 Hadamard matrix.
M12 centralizes an element of order 11 in the monster group, as a result of which it acts naturally on a vertex algebra over the field with 11 elements, given as the Tate cohomology of the monster vertex algebra.
There are 11 conjugacy classes of maximal subgroups of M12, 6 occurring in automorphic pairs, as follows: The cycle shape of an element and its conjugate under an outer automorphism are related in the following way: the union of the two cycle shapes is balanced, in other words invariant under changing each n-cycle to an N/n cycle for some integer N.