Irreducible character degree divides the group order
= Irreducible character degree divides the group order
For every irreducible complex character $\chi$ of a finite group $G$,
$$
\chi(1)\mid|G|.
$$
Indeed, row orthogonality expresses $|G|/\chi(1)$ as a sum of products of <algebraic integers>:
$$
\frac{|G|}{\chi(1)}
=\sum_C\frac{|C|\chi(c)}{\chi(1)}
\overline{\chi(c)}.
$$
It is a rational algebraic integer and therefore an integer.