What is the order of the Weyl group?

Question

Let $W$ be the Weyl group of any of the classical Lie algebras $A_n,B_n,C_n,D_n$. What is $|W|$?

A naïve calculation suggests that $$ \begin{aligned} A_n&\colon\ |W|=(n+1)!\\ B_n&\colon\ |W|=2^nn!\\ C_n&\colon\ |W|=2^nn!\\ D_n&\colon\ |W|=2^{n-1}n! \end{aligned} $$ but I haven't been able to find these expressions anywhere, so I could use the confirmation.

Answer 1

Here's a table I found on Wikipedia:

https://en.wikipedia.org/wiki/Coxeter_group#Properties

Answer 2

Not only the orders of these Weyl groups are well known, but also the Weyl groups itself:

$$ W(A_n)=S_{n+1},\; W(B_n)=W(C_n)=\Bbb{Z}_2^n\ltimes S_n, W(D_n)=\Bbb{Z}_2^{n-1}\ltimes S_n. $$

The exceptional Weyl groups are also known, but more difficult to describe. $W(G_2)$ is isomorphic to the dihedral group $D_6$ of order $12$. $W(F_4)$ is a soluble group of order $1152$, and is isomorphic to the orthogonal group $O_4(3)$ leaving invariant a quadratic form of maximal index in a $4$-dimensional vector space over the field $\Bbb{F}_3$. For $W(E_6),W(E_7)$ see here, and for $W(E_8)$ see here.