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