---
tag: 4E4GF0RJVT
kind: concept
name: Hyperoctahedral group
aliases: [hyper-octahedral-group, hyper-pyritohedral-group, hyperoctahedral-symmetry, hyperpyritohedral-group, hyperpyritohedral-symmetry, signed-permutation-group, signed-symmetric-group]
sources:
  - wikipedia: en:24268961@r1370716862
    locator: Hyperoctahedral group
    note: adopted
    mention: wikipedia://en/24268961
    sections: [Summary]
same_as:
  wikidata: Q5958362
  wikipedia: Hyperoctahedral group
inventory: [G0K5WTYRK4]
---

## Summary

The **hyperoctahedral groups** are a family of mathematical groups that arise as the [[group of symmetries|0ZG3Z5JZ85]] of the square, the cube, and their higher-dimensional counterparts (the hypercubes), as well as the corresponding dual polytopes (the regular octahedron and its higher-dimensional counterparts, the cross-polytopes). There is one hyperoctahedral group for each dimension $n$.