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Using a generalization of Cayley's hyperdeterminant as a new measure oftripartite fermionic entanglement we obtain the SLOCC classification ofthree-fermion systems with six single particle states. A special subclass ofsuch three-fermion systems is shown to have the same properties as thewell-known three-qubit ones. Our results can be presented in a unified wayusing Freudenthal triple systems based on cubic Jordan algebras. For systemswith an arbitrary number of fermions and single particle states we propose thePl\"ucker relations as a sufficient and necessary condition of separability.

Three fermions with six single particle states can be entangled in two inequivalent ways