Degree of irreducible representation need not divide 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 $$\varphi$$ does not divide the order of the fact about::derived subgroup of $$G$$.

This is a numerical non-constraint on the fact about::degrees of irreducible representations.

Related facts
For more facts, see degrees of irreducible representations.

Similar facts

 * Degree of irreducible representation may be greater than 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

Example of symmetric group of degree three
We consider the group symmetric group:S3. The standard representation of symmetric group:S3 is a faithful irreducible representation of degree two, and the derived subgroup, A3 in S3, has order three.

Example of extraspecial group
Let $$p$$ be a prime number. 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 $$p^2$$.