Square of Schur index of irreducible character in characteristic zero divides order

Statement

Suppose $G$ is a finite group, $\varphi$ is an Irreducible linear representation (?) of $G$ over $\mathbb {C}$, and $m(\chi )$ is the Schur index (?) of $\chi$. Then, $(m(\chi ))^{2}$ divides the order of $G$.

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

Related facts

• Schur index of irreducible character in characteristic zero divides exponent
• Schur index divides degree of irreducible representation