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For each finite primitive subgroup G of PGL 4 (C) , we find all the smooth G -invariant quartic surfaces. We also find all the faithful representations in PGL 4 (C) of the smooth quartic G -invariant surfaces by the groups: A 5 , S 5 ,PSL 2 (F 7) , A 6 , Z 2 4 ⋊ Z 5 and Z 2 4 ⋊ D 10. The primitive representation of these groups is precisely the subgroups of PGL 4 (C) for which P 3 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 { x 0 4 + x 1 4 + x 2 4 + x 3 4 + 12 x 0 x 1 x 2 x 3 = 0 } (unique up to projective equivalence). [ABSTRACT FROM AUTHOR] more...