Let $W$ be a finite Coxeter group. Let $$ N_W=\operatorname{max}_{g\in W}\operatorname{ord}(g) $$ where $\operatorname{ord}(g)$ denotes the order of an element $g$. By Fermat's little theorem, we know that $N_W$ divides the order of $W$. What else can we say about $N_W$? Is $N_W$ related to the Coxeter number of $W$?

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