transitively on the $W_j$, then we say $G$ is imprimitive. In this case, each $W_j$ has the same dimension, and the standard representation is induced from a representation on $W_1$. Otherwise, call $G$ primitive.

Let $A \mapsto \bar A$ denote the quotient map from $\operatorname{GL}_n({\mathbb C})$ to $\operatorname{PGL}_n({\mathbb C})$. Similarly, if $G \subset \operatorname{GL}_n({\mathbb C})$, let $\bar G$ be the image of $G$ under this map. We call the projective image $\bar G$ irreducible or primitive if $G$ is. Finite subgroups of $\operatorname{PGL}_n({\mathbb C})$ have been classified for small $n$, and we can use this to describe the finite subgroups of $\operatorname{GL}_n({\mathbb C})$.

Namely, suppose $G \subset \operatorname{GL}_n({\mathbb C})$ is irreducible. Then $Z(G)$ is a cyclic subgroup of scalar matrices, and $\bar G = G/Z(G)$. Hence the irreducible finite subgroups of $\operatorname{GL}_n({\mathbb C})$, up to isomorphism, are a subset of the set of finite cyclic central extensions of the irreducible subgroups $\bar G$ of $\operatorname{PGL}_n({\mathbb C})$.

Let $H$ be an irreducible subgroup of $\operatorname{PGL}_n({\mathbb C})$. Given one cyclic central extension $G$ of $H$ which embeds (irreducibly) in $\operatorname{GL}_n({\mathbb C})$, note that the central product $G \times_{Z(G)} C_m$ also does for any cyclic group $C_m \supset Z(G)$, and has the same 