It was constructed by Marshall Hall and David Wales (1968) as a rank 3 permutation group on 100 points.
Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family.
It is a subgroup of index two of the group of automorphisms of the Hall–Janko graph, leading to a permutation representation of degree 100.
It is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon,[1] leading to a permutation representation of degree 315.
It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w2 + w + 1 = 0, then J2 is generated by the two matrices and These matrices satisfy the equations (Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication.