The group determinant was studied during the development of representation theory. In modern language, given a finite group G, its group determinant is the representation determinant of the right regular representation of G. Let Y be a hyperbolic 3-manifold with 1 torus cusp. Suppose that Y has a finite regular cover with covering group G. We relate the group determinant to the determinant of a matrix that describes the space of non-separating surfaces with boundary in this finite cover. As a consequence we obtain that any such manifold Y has infinitely many fillings that are virtually Haken. This is joint work with Daryl Cooper.