  We study the finite dimensional complex representations of the mapping class
group $\mathcal{M}_{g,1}$ that are derived from some finite Galois coverings of
the compact oriented surface with one boundary component $\Sigma_{g,1}$. The
key ingredients are Fox derivation, Magnus modules and the Skolem-Noether
theorem, which enable us to compute the $\mathcal{M}_{g,1}$-action on the
module $L_g^{\eta}$ very explicitly, where $\eta$ is a primitive $p$the root of
unity for an odd prime $p$.
