On a congruence of Blichfeldt concerning the order of finite groups

We show that if $G$ is a finite group, $C$ a conjugacy class of $G$ and $d=\left\vert C\right\vert$, $d_{2},d_{3},\ldots,d_{m}$ are the distinct elements in the multiset $\left\{ \frac{\left\vert C\right\vert \chi(C)} {\chi(1)} \vert \chi\in \mathrm{Irr}(G)\right\}$ (here $\chi(C)$ is the value of $\chi$ on any element of $C$), then

$$\left\vert G/\left\langle C\right\rangle \right\vert \cdot\left( ... ...cdots\left( d-d_{m}\right) \equiv0 \operatorname{mod} \left\vert G\right\vert.$$

This is a dual to a generalization of a theorem of Blichfeldt stating that if $G$ is a finite group, $\theta$ a generalized character and $d=\theta (1),d_{2},d_{3},\ldots,d_{m}$ are the distinct values of $\theta$, then

$$\left\vert \ker(\theta)\right\vert \left( d-d_{2}\right) \left( d... ...dots\left( d-d_{m}\right) \equiv0 \operatorname{mod} \left\vert G\right\vert .$$

We also observe that $d=\theta(1)$ in Blichfeldt's congruence can be replaced, with a minor adjustment, by any rational value of $\theta$. A similar change can be done to the first congruence above.