Conjugation module of a group ring

The group ring $\mathbb{Z}G$ becomes a $\mathbb{Z}G$-module under the conjugation action $g\cdot x=gx{g}^{-1}$. Its basis splits into conjugacy classes, and the span of the class of $x$ is the permutation module on $G/C_G(x)$.

For a finite group $G$ and representatives $g_i$ of its conjugacy classes, Shapiro's lemma gives