A complex representation of a finite group \(G\) is a homomorphism \(\rho\colon G\to\GL(V)\) for a finite-dimensional complex vector space \(V\). Averaging an inner product on \(V\) over \(G\) gives a \(G\)-invariant inner product, for which every \(\rho(g)\) is unitary, and by Maschke’s theorem \(V\) is a direct sum of irreducible representations. The character \(\chi_V(g)=\operatorname{tr}\rho(g)\) is constant on conjugacy classes. The irreducible characters form an orthonormal basis of the space of class functions for \(\inner{\chi}{\psi}=\frac{1}{\abs{G}}\sum_{g\in G}\chi(g)\overline{\psi(g)}\), so the multiplicity of an irreducible representation \(W\) in \(V\) is \(\inner{\chi_V}{\chi_W}\). The number of irreducible characters equals the number of conjugacy classes, and \(\sum_W(\dim W)^2=\abs{G}\), the sum taken over the irreducible representations. For a subgroup \(H\le G\), induction and restriction satisfy Frobenius reciprocity, \(\inner{\operatorname{Ind}_H^G\psi}{\chi}_G=\inner{\psi}{\operatorname{Res}^G_H\chi}_H\).