The $27$-dimensional module for $E_ 6$. III

Abstract:

This is the third in a series of five papers investigating the subgroup structure of the universal Chevalley group $G = {E_6}(F)$ of type ${E_6}$ over a field $F$ and the geometry induced on the $27$-dimensional $FG$-module $V$ by the symmetric trilinear form $f$ preserved by $G$. The series uses the geometry on $V$ to describe and enumerate (up to a small list of ambiguities) all closed maximal subgroups of $G$ when $F$ is finite or algebraically closed.