The Weyl group $W_6$ of the Lie algebra $E_6$ is of order 51840, the automorphism group of the unique simple group of order 25920, while the Weyl group $W_7$ of the Lie algebra $E_7$ is of order 2903040, the direct product of the group of order 2 and the unique simple group of order 1451520.
However, it is not clear to me how the simple reflections correspond to the elements in these two groups. Also, I do not know how $W_6$ is embedded into $W_7$.
Can any expert in Lie theory and group theory help me to understand this two groups? Thanks!
Best Answer
It's important to emphasize that the simple group here is typically isomorphic to the rotation subgroup of $W$, which has index 2 and doesn't contain the reflections. So you need to look at the precise descriptions of the Weyl groups to see where the reflections fit in.
There is a lot of information in the Atlas of Finite Groups (Oxford, 1985) if you look at the appropriate entries, as well as in the exercises of Bourbaki, Groupes et algebres de Lie, Chap. VI, $\S4$ (Hermann, 1968). My textbook Reflection Groups and Coxeter Groups (Cambridge, 1990) has a summary with references in 2.12, including the Atlas notation. As the comments indicate, there are sometimes multiple descriptions of an exceptional Weyl group in terms of known finite linear groups, so it takes some sorting out. For example, an embedding of the Weyl group of type $E_6$ into the one of type $E_7$ is probably best understood in terms of the $\mathrm{Sp}_6(2)$ model, as suggested in the comments.