Degree of irreducible representation may be greater than order of derived subgroup

Statement

It is possible to have a finite group $G$ and an irreducible linear representation $\varphi$ of $G$ over a splitting field in characteristic zero such that the degree of $\varphi$ is strictly greater than the order of the Derived subgroup (?) of $G$.

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

Related facts

For more facts, see degrees of irreducible representations.

Similar facts

• Degree of irreducible representation need not divide order of derived subgroup
• Degree of irreducible representation need not divide exponent
• Degree of irreducible representation may be greater than exponent

Opposite facts

• Degree of irreducible representation divides order of group
• Degree of irreducible representation divides index of center
• Degree of irreducible representation divides index of abelian normal subgroup
• Degree of irreducible representation is bounded by index of abelian subgroup

Related facts about conjugacy class sizes

These facts are related to the conjugacy class size statistics of a finite group.

• Size of conjugacy class is bounded by order of derived subgroup
• Size of conjugacy class need not divide order of derived subgroup

Proof

Example of extraspecial group

Consider an extraspecial group of order $p^{5}$ for any prime number $p$. The derived subgroup has order $p$. However, this group has a faithful irreducible representation of degree $p^{2}$.