Lower bounds for GL₂(F₎ number fields

Let Fₙ(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 ≥ 1. In this paper, we give a lower bound for the cases G=GL₂(F₎, \; PGL₂(F₎ for primes ≥ 13. We also provide a method to compute lower bounds for any permutation representations of these groups.