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

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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.

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