Narrow Results

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