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.

Related facts

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