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

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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

Opposite facts

Related facts about conjugacy class sizes

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

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.