Petersen graph and monodromy of the 27 lines on the Clebsch surface

Let $G$ be the orbifold fundamental group of the moduli space of smooth cubic surfaces ${\mathcal{M}}_{\mathsf{s}\mathsf{m}}$ in ${\mathbb{P}}_{\mathbb{C}}^3$ with base point at the Clebsch surface $X_{\mathbf{1}}$. The image of the monodromy action G \to \lbrace \text{Permutations of27$lineson$X_{$\mathbf{1}$}} \rbrace is famously the Weyl group of type $E_6$. Here we give a description of this monodromy action in terms of the Petersen graph by working out the action of ten explicit generators of $G$ by elementary calculation. These ten generators were found in joint work with Allcock and Looijenga while studying the description of ${\mathcal{M}}_{\mathsf{s}\mathsf{m}}$ as a discriminant complement in a complex $4$-ball quotient.