Monomial modular representations and symmetric generation of the Harada-Norton group. (Q1415361)

The second author [in J. Algebra 184, No. 3, 1205-1227 (1996; Zbl 0858.20020)], constructed the Held group \(He\) using a 15-dimensional 7-modular monomial representation of \(3^\cdot A_7\), the triple cover of \(A_7\). He obtained \(He\) as a homomorphic image of a split extension of a free product of 15 cyclic subgroups of order 7 by the group \(3^\cdot A_7\). In this paper, the much larger Harada-Norton group \(HN\) is constructed in a similar manner. A 352-dimensional 5-modular monomial representation of \(2^\cdot HS:2\), a double cover of the automorphism group of the Higman-Sims group \(HS\), is used to build a split extension \(\mathcal P\) of a free product of 352 cyclic subgroups of order 5 by the group \(2^\cdot HS:2\). The group \(\mathcal P\) has \(HN\) as a ``natural'' homomorphic image. An outline of how this approach can be used to construct the 133-dimensional representation of \(HN\) over \(\mathbb{Q}(\sqrt 5)\), which is the smallest degree of a ``true'' characteristic 0 representation of \(\mathcal P\), is given. As in the Held case, extension to the automorphism group of \(HN\) follows easily.