# Construction of representations

Is there an example, where given a conjugacy class in a finite group, can we construct an irreducible representation from it?

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For most finite groups, the bijection between conjugacy classes and irreducible $\mathbb{C}$-representations is not in any sense (known to me, at least) canonical. One sees this already for finite abelian groups: the isomorphism between such a group $G$ and its character group is non-canonical, and this at least can be formalized and proved using category theory.
Why can the embedding of a finite group into $S_n$ not be used here? –  plusepsilon.de Jun 9 '11 at 6:35