Let $\gamma$ be an $n$-dimensional complex representation of a finite group $G$ with character $\chi$ and let $e=c\_0, c\_1, ..., c\_{\ell}$ be a set of conjugacy class representatives for $G$. In the case where $\gamma$ is faithful, I recently obtained the formula $$ \frac{1}{|G|} \prod\_{i=1}^{\ell} (n-\chi(c\_i))$$ as the size of a certain abelian group associated with $\gamma$ (see Theorem 3 in [this paper](https://arxiv.org/abs/1606.00798) if interested). In particular, this implies that this quantity is always an integer (if $\gamma$ is not faithful, then the product is 0). 


Is there a nice way to see that this is an integer just using basic character theory?



Why is this character expression an integer?
Yes, this is an observation of H. Blichfeldt ( I think J-P. Serre also attributes it to Minkowski), which has been rediscovered many times over the years. Letting $1$ denote the trivial character, note that 
$\theta = \prod\_{i = 1}^{\ell} ( \chi - \chi(c\_{i})1) $ is an algebraic integer combination of characters of $G$, and also vanishes everywhere on $G$ except the identity, where its value is $\prod\_{i = 1}^{\ell} ( n - \chi(c\_{i})) .$ Hence $\theta$ is an algebraic integer multiple of the regular character of $G$, and 
$\langle \theta,1 \rangle = \frac{1}{|G|}\prod\_{i = 1}^{\ell} ( n - \chi(c\_{i})) $
is an algebraic integer ( which is easily seen to be rational by Galois theory, hence is a rational integer).


(Edit: In fact, it is only necessary to take the distinct values of $\chi(c\_{i})$ without repetition in the product which is what Blichfeldt did).


