  Consider the family of smooth cubic surfaces which can be realized as
threefold-branched covers of $\mathbb{P}^{2}$, with branch locus equal to a
smooth cubic curve. This family is parametrized by the space $\mathcal{U}_{3}$
of smooth cubic curves in $\mathbb{P}^{2}$ and each surface is equipped with a
$\mathbb{Z}/3\mathbb{Z}$ deck group action.
  We compute the image of the monodromy map $\rho$ induced by the action of
$\pi_{1}\left(\mathcal{U}_{3}\right)$ on the $27$ lines contained on the cubic
surfaces of this family. Due to a classical result, this image is contained in
the Weyl group $W\left(E_{6}\right)$. Our main result is that $\rho$ is
surjective onto the centralizer of the image a of a generator of the deck
group. Our proof is mainly computational, and relies on the relation between
the $9$ inflection points in a cubic curve and the $27$ lines contained in the
cubic surface branching over it.
