What is the structure of the Coxeter groups of type $\text{D}_n$

Question

I am curious on the structure of the Coxeter group $G$ of type $\text{D}_n$. Here I let $\{e_1,\cdots,e_n\}$ be the standard basis of the vector space $\mathbb{R}^n$. Then I choose $$r_k=e_{k+1}-e_k~\mbox{for}~1\leqslant k<n~~~\mbox{and}~~~r_n=e_1+e_2$$to be the simple roots of $G$. So we will have the root system $\{e_i\pm e_j \mid i,j=1,\cdots,n~~\mbox{and}~~i\neq j\}$. Denote the reflection of $r_k$ by $\sigma_k$, and the reflection of $e_i+e_j$ by $f_{i,j}$. It is clear that $H:=\langle\sigma_1,\cdots,\sigma_{n-1}\rangle\cong\text{S}_n$ the symmetric group as it represents all the permutations of the standard basis $\{e_1,\cdots,e_n\}$. Then $\sigma_kf_{i,j}\sigma_k^{-1}=f_{i',j'}$ where $i',j'$ are the resulting elements of $i,j$ under the induced permutation of $\sigma_k$. Hence, we can think of $H$ acting on the set $\{f_{i,j} \mid \mbox{all possible}~i,j\}$. But then what is the next to deduce the explicit structure of $G$?