In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube.
As a Coxeter group it is of type Bn = Cn, and as a Weyl group it is associated to the orthogonal groups in odd dimensions. As a wreath product it is where is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations π either of the set { −n, −n + 1, ..., −1, 1, 2...