Cubic fourfolds admitting a cyclic group of symplectic automorphisms of order not a power of 2 are rational and lie in the Hassett divisors C_14 or C_42.
Fu, Classification of polarized symplectic autormorphisms of Fano varieties of cubic fourfolds , Glasgow Mathematical Journal, Vol
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Rational cubic fourfolds with a symplectic group of automorphisms
Cubic fourfolds admitting a cyclic group of symplectic automorphisms of order not a power of 2 are rational and lie in the Hassett divisors C_14 or C_42.