Cubic fourfolds with symplectic automorphisms

We determine projective equations of smooth complex cubic fourfolds with symplectic automorphisms by classifying 6-dimensional projective representations of Laza and Zheng’s 34 groups. In particular, we determine the number of irreducible components for moduli spaces of cubic fourfolds with symplectic actions by these groups. We also discuss the fields of definition of cubic fourfolds in six maximal cases.