There is a particular method in Reflection groups and Coxeter groups by Humphreys to compute the orders of various irreducible finite Coxeter groups in Chapter 2.11. The method involves using group action of a finite group $G$ acting as a permutation group on a set $X \ni x$ so that $|G|=|Gx||G_x|$, where $Gx$ is the orbit of $x$ and $G_x$ is the isotropy group of $x$.
In the context of irreducible finite Coxeter groups $W$, $W$ acts on the root system $\Phi$. The main exercise in that section asks to compute the orders of groups of type $\mathrm{A}_n$, $\mathrm{B}_n$, and $\mathrm{D}_n$ ($\mathrm{A}_n$ is not the alternating group of order $n!/2$ and $\mathrm{D}_n$ is not the dihedral group of order $2n$).
My question is, how can we figure out the orbits and isotropy groups for the groups in the exercise? I'm finding it trickier to do this for the groups of type $\mathrm{B}_n$ and $\mathrm{D}_n$. Also, even though I know that the group of type $\mathrm{A}_n$ is the symmetric group $S_{n+1}$ of order $(n+1)!$, I'm not entirely sure how to end up with this order using the above suggested method. Are there any good references I could use?