Hurwitz generation in groups of types F4, E6 2E6, E7 and E8

A Hurwitz generating triple for a group G is an ordered triple of elements (x, y, z) ∈G3 where x2 = y3 = z7 = xyz = 1 and (x, y, z) = G. For the finite quasisimple exceptional groups of types F4, E6, 2E6, E7 and E8, we provide restrictions on which conjugacy classes x,y and z can belong to if (x, y, z) is a Hurwitz generating triple. We prove that there exist Hurwitz generating triples for F4(3), F4(5), F4(7), F4(8), E6(3) and E7 (2), and that there are no such triples for F4 (23n -2), F4 (23n - 1), E6 (73n - 2), E6 (73n - 1), SE6 (7n) or 2E6(7n) when n ≥ 1.