= Conjugacy-class formula for an induced character
{title2=$\operatorname{Ind}_H^G\chi(g)=\frac{|C_G(g)|}{|H|}\sum_{h\in g^G\cap H}\chi(h)$}
For a <finite group> $G$, a <subgroup> $H$, and a <character> $\chi$ of $H$, each element $h$ of $g^G\cap H$ has exactly $|C_G(g)|$ preimages under $x\mapsto x^{-1}gx$. This follows because one such preimage $x_0$ has full fiber $C_G(g)x_0$. Grouping the <character formula for an induced representation> by these fibers proves the formula. Grouping again by <conjugacy classes> of $H$ replaces each class by its size times its <character> value.
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