Let M denote the moduli space of smooth cubic threefolds. We prove that the monodromy 1(M) ! Sp(10;Z) of the 3rd cohomology groups on the fibres of the universal family does not factor through the genus five mapping class group. This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds, and provides evidence for a corresponding symplectic nonrationality conjecture.