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Consider the family of smooth cubic surfaces which can be realized asthreefold-branched covers of $\mathbb{P}^{2}$, with branch locus equal to asmooth 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 cubicsurfaces of this family. Due to a classical result, this image is contained inthe Weyl group $W\left(E_{6}\right)$. Our main result is that $\rho$ issurjective onto the centralizer of the image a of a generator of the deckgroup. Our proof is mainly computational, and relies on the relation betweenthe $9$ inflection points in a cubic curve and the $27$ lines contained in thecubic surface branching over it.

Monodromy of the family of cubic surfaces branching over smooth cubic curves