Degree of irreducible representation may be greater than exponent

Statement

We can have a finite group and an irreducible linear representation of the group over an algebraically closed field of characteristic zero, such that the degree of the irreducible representation is greater than the exponent of the group.

This is a non-constraint on the Degrees of irreducible representations (?) of a finite group.

Related facts

Similar facts

• Degree of irreducible representation need not divide exponent
• Square of degree of irreducible representation need not divide order
• Size of conjugacy class need not divide exponent

Opposite facts

• Schur index of irreducible character in characteristic zero divides exponent, Schur index divides degree of irreducible representation: Thus, the Schur index of an irreducible character/representation divides both the degree of the representation and the exponent.
• Degree of irreducible representation divides group order
• Degree of irreducible representation divides order of inner automorphism group (i.e., the degree divides the index of the center)
• Degree of irreducible representation divides index of abelian normal subgroup
• Order of inner automorphism group bounds square of degree of irreducible representation

Proof

Example of big extraspecial groups

Consider an extraspecial group of order $p^{7}$ for any prime $p$. The exponent of this group is either $p$ or $p^{2}$. On the other hand, it admits a faithful irreducible representation of order $p^{3}$.