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  "aliases": [
    "hyper-octahedral-group",
    "hyper-pyritohedral-group",
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    "hyperpyritohedral-group",
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    "signed-permutation-group",
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  "body": "\n## Summary\n\nThe **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 $n$.\n",
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