Asymptotic Bounds for the Size of Hom$(A,{\mathrm{G}\mathrm{L}}_n(q))$

Fix an arbitrary finite group $A$ of order $a$, and let $X(n,q)$ denote the set of homomorphisms from $A$ to the finite general linear group ${\rm GL}_n(q)$\mathrm{.}Thesizeof$X(n,q)$isapolynomialin$q$. In this note it is shown that generically this polynomial has degree $n^2(1-a^{-1})-{ϵ}_r$ and leading coefficient $m_r$, where ${ϵ}_r$ and $m_r$ are constants depending only on $r:=n\mathrm{m}\mathrm{o}\mathrm{d}a$. We also present an algorithm for explicitly determining these constants.