Hurwitz Generation of PSp6(q)

We show that the symplectic groups PSp6(q) are Hurwitz for all q = p^m ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp6(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over F_{p^m}, contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9] and [10].