J1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.
J1 is the unique finite group G with the property that for C any nontrivial conjugacy class, every element of G is equal to xy for some x, y in C.[3] Janko found a modular representation in terms of 7 × 7 orthogonal matrices in the field of eleven elements, with generators given by and Y has order 7 and Z has order 5.
This permutation representation can be constructed implicitly by starting with the subgroup PSL2(11) and adjoining 11 involutions t0,...,tX.
PSL2(11) permutes these involutions under the exceptional 11-point representation, so they may be identified with points in the Payley biplane.
Janko (1966) found the 7 conjugacy classes of maximal subgroups of J1 shown in the table.