Monomial modular representations and symmetric generation of the Harada–Norton group

AbstractThis paper is a sequel to Curtis [J. Algebra 184 (1996) 1205–1227], where the Held group was constructed using a 7-modular monomial representation of 3·A7, the exceptional triple cover of the alternating group A7. In this paper, a 5-modular monomial representation of 2·HS:2, a double cover of the automorphism group of the Higman–Sims group, is used to build an infinite semi-direct product P which has HN, the Harada–Norton group, as a ‘natural’ image. This approach assists us in constructing a 133-dimensional representation of HN over Q(5), which is the smallest degree of a ‘true’ characteristic 0 representation of P. Thus an investigation of the low degree representations of P produces HN. As in the Held case, extension to the automorphism group of HN follows easily