Weyl group of type $E_6$ and K3 surfaces

Adapting methods of previous papers by A. Sarti and the author, we construct K3 surfaces from invariants of the Weyl group of type ${\text{Erm}}_6$. We study in details one of these surfaces, which turns out to have Picard number $20$: for this example, we describe an elliptic fibration (and its singular fibers), the Picard lattice and the transcendental lattice.