  We apply representation theory to study the homology of equivariant
Dehn-fillings of a given finite, regular cover of a compact 3-manifold with
boundary a torus. This yields a polynomial which gives the rank of the part of
the homology carried by the solid tori used for Dehn-filling. The polynomial is
a symmetrized form of the group determinant studied by Frobenius and Dedekind.
As a corollary every such hyperbolic 3-manifold has infinitely many virtually
Haken Dehn-fillings.
