Monomial Modular Representations and Construction of the Held Group

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Monomial representations of familiar finite groups over finite fields are used to construct (infinite) semi-direct products of free products of cyclic groups by groups of monomial automorphisms. Finite homomorphic images of theseprogenitorsin which the actions on the group of automorphisms and on the cyclic components are faithful are sought. The smallest non-trivial images of this type are often sporadic simple groups. The technique is demonstrated by three examples over the fieldsZ3,Z5, andZ7, which produce the Mathieu groupM11, the unitary groupU3(5):2, and the Held group, respectively.