with a normal subgroup $N$ such that $H/N \simeq G$. The extension is called central or cyclic if $N$ is a central or cyclic subgroup of $H$.

If $G$, $H$, and $Z$ are groups such that $Z \subset Z(G) \cap
Z(H)$, then the central product $G \times_Z H$ of $G$ and $H$ with respect to $Z$ is defined to be direct product $G \times H$ modulo the central subgroup $\{ (z, z) : z \in Z \}$.

If $\chi_1, \chi_2$ are characters of $G$, their inner product is $(\chi_1, \chi_2) = |G|^{-1} \sum_G \chi_1(g) 
\overline{\chi_2(g)}$.

We denote a cyclic group of order $m$ by $C_m$.

Finite subgroups of $\operatorname{GL}_n({\mathbb C})$
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Next we recall some definitions and facts about finite subgroups of $\operatorname{GL}_n({\mathbb C})$.

Let $G$ be a finite subgroup of $\operatorname{GL}_n({\mathbb C})$, so one has the standard representation of $G$ on $V = {\mathbb C}^n$. We say $G$ is reducible if there exists a nonzero proper subspace $W \subset V$ which is fixed by $G$.

Suppose $G$ is irreducible. Schur’s lemma implies that $Z(G) \subset Z(\operatorname{GL}_n({\mathbb C}))$. In particular, $Z(G)$ is cyclic. If there exists a nontrivial decomposition $V = W_1 \oplus \cdots \oplus W_k$ such that $G$ acts 