It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally.
It has a presentation in terms of three generators a, b, and c as Alternatively, one can start with the subgroup M24 and adjoin 3975 involutions, which are identified with the trios.
By adding a certain relation, certain products of commuting involutions generate the binary Golay cocode, which extends to the maximal subgroup 211:M24.
Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J4.
Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below.