  We introduce invariants of Hurwitz equivalence classes with respect to
arbitrary group $G$. The invariants are constructed from any right $G$-modules
$M$ and any $G$-invariant bilinear function on $M$, and are of bilinear forms.
For instance, when $G$ is the mapping class group of the closed surface,
$\mathcal{M}_g$, we get an invariant of 4-dimensional Lefschetz fibrations over
the 2-sphere. Moreover, the construction is applicable for the quantum
representations of $\mathcal{M}_g $ derived from Chern-Simons field theory. We
compute the associated invariants in some cases, and find infinitely many
Lefschetz fibrations which have the same Seiberg-Witten invariant and are
homeomorphic but not mutually isomorphic as fibrations.
