Weyl group of Dn ($W(D_n)\cong (\mathbb{Z}/2\mathbb{Z}{)}^{n-1}⋊S_n$)

The Weyl group of the Dn root system consists of coordinate permutations together with an even number of sign changes. Reflection in ${\epsilon }_i-{\epsilon }_j$ swaps coordinates $i,j$; reflection in ${\epsilon }_i+{\epsilon }_j$ swaps them and negates both. These generate all even signed permutations, giving order $2^{n-1}n!$.