Degree of irreducible projective representation divides order of group

Statement
Suppose $$G$$ is a finite group, $$K$$ is an algebraically closed field whose characteristic does not divide the order of $$G$$, and $$\varphi$$ is an irreducible projective representation of $$G$$ over $$K$$ of degree $$d$$. Then, $$d$$ divides the order of $$G$$.

This gives a numerical constraint on the degrees of irreducible projective representations.

Stronger facts

 * Degree of irreducible projective representation divides index of cyclic normal subgroup
 * Degree of irreducible projective representation divides index of abelian normal subgroup to which its cohomology class restricts trivially

Case of ordinary (linear) representations

 * Degree of irreducible representation divides order of group