Irreducible Monomial Linear Groups of Degree Four over Finite Fields

We describe an algorithm for explicitly listing the irreducible monomial subgroups of GL(n, q) , given a suitable list of finite irreducible monomial subgroups of GL(n, C) , where n is 4 or a prime, and q is a prime power. Particular attention is paid to the case n = 4, and the algorithm is illustrated for n = 4 and q = 5. Certain primitive permutation groups can be constructed from a list of irreducible monomial subgroups of GL(n, q) . The paper’s final section shows that the computation of automorphisms of such permutation groups reduces mainly to computation of irreducible monomial subgroups of GL(n, q) , q prime.