Hurwitz Generation in Groups of Types $F_4$, $E_6$, $^2E_6$, $E_7$ and $E_8$.

A Hurwitz generating triple for a group $G$ is an ordered triple of elements $(x,y,z) \in G^3$ where $x^2=y^3=z^7=xyz=1$ and $\langle x,y,z \rangle = G$. For the finite quasisimple exceptional groups of types $F_4$, $E_6$, $^2E_6$, $E_7$ and $E_8$, 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 $F_4(3)$, $F_4(5)$, $F_4(7)$, $F_4(8)$, $E_6(3)$ and $E_7(2)$, and that there are no such triples for $F_4(2^{3n-1})$, $F_4(2^{3n-2})$, $SE_6(7^n)$ and ${}^2E_6(7^n) \cong {}^2SE_6(7^n)$, when $n \geq 1$.