Signed symmetric group ($S{S}_n$)

The signed symmetric group consists of permutations of $\{\pm 1,\dots ,\pm n\}$ commuting with sign change. It is the Coxeter group of type $B_n$.

The even signed symmetric group is the index-two subgroup of signed permutations with an even number of sign changes. It is the Coxeter group of type $D_n$.