Let $G$ a finite group which act on a finite set $M$. Let $C(M) : = Map(M, \mathbb{C}:= \{f: M \rightarrow \mathbb{C} \}$ the vector space of complex values functions from $M$. The group $G$ act on $C(M)$ via \begin{align} G \times C(M) \rightarrow C(M), \, \, (g, f) \mapsto g(f): M \rightarrow \mathbb{C}, \, \, m \mapsto f(g^{-1}(m)) \end{align} Show that $C(M)$ is a representation of $G$ and for the character $\chi_{C(M)}$ we have that $\chi_{C(M)}(g) = |M^{G}| \, \, \, \forall \, g\in G$, where $M^{G}:= \{m\in M | g(m) = m \, \, \, \forall g\in G \}$ is the set of fixed points in $M$ for the $G$-action.
I showed that $C(M)$ is a representation of $G$, but I have some problem with the proof of $$\chi_{C(M)}(g) = |M^{G}| \, \, \, \forall \, g\in G$$ Any suggestions? Thanks in advance!