Schur index of irreducible character in characteristic zero divides exponent

Statement
Suppose $$G$$ is a finite group, $$\varphi$$ is an fact about::irreducible linear representation of $$G$$ over $$\mathbb{C}$$, and $$m(\chi)$$ is the fact about::Schur index of $$\chi$$. Then, $$m(\chi)$$ divides the exponent of $$G$$.

Here, $$\mathbb{C}$$ can be replaced by any splitting field for $$G$$ of characteristic zero.

Related facts

 * Square of Schur index of irreducible character in characteristic zero divides order
 * Schur index divides degree of irreducible representation