Degree of irreducible representation divides index of center

Statement
Let $$G$$ be a finite group and $$k$$ be an algebraically closed field of characteristic zero. Let $$\rho$$ be an irreducible representation of $$G$$ over $$k$$. Then, the degree of $$\rho$$ divides the index $$[G:Z(G)]$$ of the center $$Z(G)$$ in $$G$$. (Note that since the quotient by the center equals the inner automorphism group, this is equivalent to saying that the degree of $$\rho$$ divides the order of the inner automorphism group).

Similar facts about degrees of irreducible representations

 * Degree of irreducible representation divides group order
 * Order of inner automorphism group bounds square of degree of irreducible representation
 * Degree of irreducible representation divides index of abelian normal subgroup
 * Sum of squares of degrees of irreducible representations equals order of group

Analogous facts about size of conjugacy class

 * Size of conjugacy class divides order of inner automorphism group
 * Size of conjugacy class divides order of group