Invariant smooth quartic surfaces by all finite primitive subgroups of [formula omitted

Abstract

For each finite primitive subgroup G of PGL4(C), we find all the smooth G-invariant quartic surfaces. We also find all the faithful representations in PGL4(C) of the smooth quartic G-invariant surfaces by the groups: A5,S5,PSL2(F7), A6, Z24⋊Z5 and Z24⋊D10. The primitive representation of these groups is precisely the subgroups of PGL4(C) for which P3 is not G-super rigid. As a byproduct, we show that the smooth quartic surface with the biggest group of projective automorphism is given by {x04+x14+x24+x34+12x0x1x2x3=0} (unique up to projective equivalence).