4
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The Weyl Group of $F_4$ is of order $1152=2^{7} \cdot 3^{2}$. By Burnside's theorem the group is solvable.

Is there a way to see solvability from the root system? Is it possible to see the order of the group there?

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    @Beginner: It is true that the long roots in $F_4$ form a root system of type $D_4$. But there are also copies of $B_4$ in $F_4$. They consist of all the long roots and additionally 8 of the short roots, as Jyrki wrote. Actually, $F_4$ can be realised as the vectors $\pm e_i$ ($1 \le i \le 4$, gives 8 short vectors), $\pm e_i \pm e_j$ (1\le i < j \le 4, gives 24 long vectors) and $\frac{1}{2}(e_1 \pm e_2 \pm e_3 \pm e_4)$ (gives 16 more short vectors) in $\mathbb{R}^4$. The first 30 vectors there are the standard realisation of $B_4$.2017-02-03

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