Construction of Symmetric Cubic Surfaces

We consider the action of the group ${\mathrm{P}\mathrm{G}\mathrm{L}}_4(K)$ on the smooth cubic surfaces of ${\mathbb{P}}_K^3$ ($K$ an algebraically closed field of characteristic zero). We classify, in an explicit way, all the smooth cubic surfaces with non trivial stabilizer, the corresponding stabilizers and obtain a geometric description of each group in terms of permutations of the Eckardt points, of the $27$ lines or of the $45$ tritangent planes.