  We consider the action of the group $\mathrm{PGL}_4(K)$ on the smooth cubic
surfaces of $\mathbb{P}^3_K$ ($K$ 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.
