Based on this question (which focuses on the case $E_8$) I wonder the following:

Question:For each finite reflection group $\Gamma\subseteq\mathrm O(\Bbb R^d)$, what is the largest finite group $\bar \Gamma\subseteq\mathrm O(\Bbb R^d)$ that contains $\Gamma$?

Some inclusions already happen among the reflection groups and their extensions:

- For $I_2(n)$ there is no such largest group, because $$I_2(n)\subset I_2(2n)\subseteq I_2(4n)\subset\cdots\subset I_2(2^r n)\subset \cdots.$$
- In general, we have $D_d\subset B_d$.
- In general, $A_d\subset A_d^*$, where $A_d^*$ is the extended group that results from the additional symmetries of the Coxeter-Dynkin diagram of $A_d$.
- We have $A_3=D_3\subset B_3=A_3^*$.
- We have $A_4\subset A_4^*\subset H_4$.
- We have $D_4\subset B_4\subset F_4\subset F_4^*$, again, $F_4^*$ is the extended group resulting from the additional symmetries of the Coxeter-Dynkin diagram.
- We have $E_6\subset E_6^*$, for the same reason as above.
- We have $A_7\subset A_7^*\subset E_7$.
- We have $A_8\subset A_8^*\subset E_8$.

(Thanks to Daniel for noting the extensions of $A_4^*, A_7^*$ and $A_8^*$).

I believe that $H_3, H_4,B_5,E_7,E_8$ and $B_d,d\ge 9$ cannot be enlarged by the same reasoning as given in this answer (because these groups are the largest reflection groups in their respective dimension and their Coxeter-Dynkin diagrams have no additional symmetries).

So we are left with the following:

Question:Can we enlarge the groups $B_d$ ($d\in\{3,5,6,7\}$), $F_4^*$, $E_6^*$ and $A_d^*$ ($d\not\in\{4,7,8\}$)?

Maybe the inclusion $A_d\subset A_d^*\subset\cdots$ is not the right chain leading to the largest group. And I also have not touched on the reducible groups which can also be enlarged in some cases, e.g. the Coxeter-Dynkin diagram of $I_2(n)\oplus I_2(n)$ has also additional symmetries. We also have inclusions like $I_1\oplus B_d \subset B_{d+1}$.

10more comments