Hyperoctahedral

Hyperoctahedral is an adjective used in geometry and group theory to describe objects and symmetries associated with the hyperoctahedron, hypercube, or their higher-dimensional generalizations. In common usage it refers to the symmetry group and related structures in n dimensions, often denoted Bn or Cn in the context of Lie theory and Coxeter groups.

In group-theoretic terms, the hyperoctahedral group Bn (also called Cn in some conventions) is the symmetry group of the n-dimensional hypercube and the n-dimensional cross-polytope. It can be described as the wreath product Z2 wr Sn, and has order 2^n n!. Equivalently, it consists of all signed permutation matrices: matrices with exactly one nonzero entry in each row and column, where the nonzero entries are ±1. This makes Bn isomorphic to the semidirect product (Zn2)^n ⋊ Sn, reflecting independent bitwise sign changes together with coordinate permutations.

The hyperoctahedral group acts on R^n by permuting coordinates and flipping signs, preserving the lattice Z^n. It is a Weyl group of type Bn (and also type Cn in another labeling), and serves as the full symmetry group of both the n-cube (hypercube) and the n-orthoplex (cross-polytope). The orientation-preserving subgroup has index 2 and order 2^(n−1) n!.

Objects with hyperoctahedral symmetry are sometimes described as hyperoctahedral polytopes. In geometry and combinatorics, the term highlights the rich symmetry structure that generalizes square and cube symmetries to higher dimensions.