  For a finite group $G$ we investigate the smallest positive integer $e(G)$
such that the map sending $g\in G$ to $e(G)|G:C_G(g)|$ is a generalized
character of $G$. It turns out that $e(G)$ is strongly influenced by local
data, but behaves irregularly for non-abelian simple groups. We interpret
$e(G)$ as an elementary divisor of a certain non-negative integral matrix
related to the character table of $G$. Our methods applied to Brauer characters
also answers a recent question of Navarro: The $p$-Brauer character table of
$G$ determines $|G|_{p'}$.
