Here's an example that involves a group other than the symmetric group.
Let ${\mathbb F}_{q^n}$ denote the finite field with $q^n$ elements. Since ${\mathbb F}_{q^n}$ is a vector space of dimension $n$ over ${\mathbb F}_q$, we may regard a generator of the multiplicative group ${\mathbb F}^*_{q^n}$ as an element of $GL_n({\mathbb F}_q)$; define a Singer cycle to be such a generator. Also define a reflection to be a nontrivial element of $GL_n({\mathbb F}_q)$ whose fixed points form a hyperplane.
Theorem (Lewis–Reiner–Stanton). The number of factorizations of a Singer cycle into a product of $n$ reflections is $(q^n-1)^{n-1}$.
The only known proof uses the character theory of $GL_n({\mathbb F}_q)$ in an essential way.
Note that the above theorem is analogous to the classical result that the number of factorizations of an $n$-cycle in ${\mathfrak S}_n$ into a product of $n-1$ transpositions is $n^{n-2}$.
