Degree of irreducible representation need not divide exponent

Statement
We can have a finite group and an irreducible linear representation of the group over an algebraically closed field of characteristic zero (or more generally over a splitting field), such that the degree of the irreducible representation does not divide the exponent of the group.

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

Similar facts

 * Degree of irreducible representation may be greater than 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

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 degree $$p^3$$.

For odd primes $$p$$, we can take an example of an extraspecial group of order $$p^5$$ and exponent $$p$$, which admits a faithful irreducible representation of degree $$p^2$$.

Example of symmetric group of degree six
The symmetric group of degree six has an irreducible representation of degree $$16$$ over the rational numbers, arising from the self-conjugate partition $$6 = 3 + 2 + 1$$. On the other hand, the exponent of the group is the lcm of the numbers from $$1$$ to $$6$$, which is $$60$$, and is not a multiple of $$16$$.