Let $G$ be a finite group, and let $\operatorname{Cl}(G)$ denote the set of conjugacy classes of $G$. Define the complex [vector space](/page/Vector%20Space) of class functions on $G$ by
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\begin{align*}
\operatorname{CF}(G) := \{f: G \to \mathbb{C} \mid f(hgh^{-1}) = f(g) \text{ for all } g,h \in G\}.
\end{align*}