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 $x^{2} = y^{3} = z^{7} = x y z = 1$ and $⟨ x, y, z ⟩ = G$ . For the finite quasisimple exceptional groups of types $F_{4}$ , $E_{6}$ , $^{2} E_{6}$ , $E_{7}$ 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 $F_{4} (2^{3 n - 2})$ , $F_{4} (2^{3 n - 1})$ , $E_{6} (7^{3 n - 2})$ , $E_{6} (7^{3 n - 1})$ , $S E_{6} (7^{n})$ or $^{2} E_{6} (7^{n})$ when $n \geq 1$ .

Hurwitz Generation in Groups of Types $F_{4}$ , $E_{6}$ , $^{2} E_{6}$ , $E_{7}$ and $E_{8}$

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Hurwitz Generation in Groups of Types $F_{4}$ , $E_{6}$ , $^{2} E_{6}$ , $E_{7}$ and $E_{8}$

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Hurwitz Generation in Groups of Types F4F4​, E6E6​, 2E62E6​, E7E7​ and E8E8​

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Hurwitz Generation in Groups of Types F4F4​, E6E6​, 2E62E6​, E7E7​ and E8E8​

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Hurwitz Generation in Groups of Types $F_{4}$ , $E_{6}$ , $^{2} E_{6}$ , $E_{7}$ and $E_{8}$

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Hurwitz Generation in Groups of Types $F_{4}$ , $E_{6}$ , $^{2} E_{6}$ , $E_{7}$ and $E_{8}$

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