Classifying finite monomial linear groups of prime degree in characteristic zero

Let p be a prime and let C be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of GL(p, C) up to conjugacy. That is, we give a complete and irredundant list of GL(p, C)-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of GL(p, C) is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For p <= 3, we classify all finite irreducible subgroups of GL(p, C). Our classifications are available publicly in Magma.