Lower bounds for ${\text{GL}}_2({\mathbb{F}}_{\mathrm{\ell }})$ number fields

Let ${\mathcal{F}}_n(X;G)$ denote the set of number fields of degree $n$ with absolute discriminant no larger than $X$ and Galois group $G$. This set is known to be finite for any finite permutation group $G$ and $X\ge 1$. In this paper, we give a lower bound for the cases $G={\text{GL}}_2({\mathbb{F}}_{\mathrm{\ell }}),{\text{PGL}}_2({\mathbb{F}}_{\mathrm{\ell }})$ for primes $\mathrm{\ell }\ge 13$. We also provide a method to compute lower bounds for any permutation representations of these groups.