On the $K3$ surface with ${\mathfrak{S}}_4\times {\mathfrak{S}}_4$ action

By a lattice theoretic approach, Brandhorst–Hashimoto has made the list of K3 surfaces with finite groups of automorphisms which properly contain a maximal symplectic automorphism group. We give $3$ different explicit descriptions to the $K3$ surface with an action of ${\mathfrak{S}}_4\times {\mathfrak{S}}_4$, with various characterizations, and construct an explicit isomorphism to the Schur’s quartic. We also calculate the intersection of the two polarization-preserving finite automorphism groups.