On overgroups of $\mathrm{E}(\mathrm{E}_6,R)$ and $\mathrm{E}(\mathrm{E}_7,R)$ in their minimal representations

The article is dedicated to description of overgroups of elementary Chevalley groups of types $\mathrm{E}_6$ and $\mathrm{E}_7$ in their minimal irreducible representations. One of the first step towards such description, namely construction of the series of perfect intermediate groups, is made. Probably all perfect intermediate groups for groups of type $\mathrm{E}_6$ are described. The hypothesis concerning the structure of perfect overgroups of type $\mathrm{E}_7$ is made.