The C2 group has order 8 as shown on this circle

The C3 (Oh) group has order 48 as shown by these spherical triangle reflection domains.

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

 of a hypercube

Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

 or of a cross-polytope

Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...

. Groups of this type are identified by a parameter n, the dimension of the hypercube.

As a Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

 it is of type B_n = C_n, and as a Weyl group

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...

 it is associated to the orthogonal group

Orthogonal group

In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

s in odd dimensions. As a wreath product

Wreath product

In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.Given two groups A and H...

 it is where is the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

 of degree n. As a permutation group

Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...

, the group is the signed symmetric group of permutations π either of the set { −n, −n + 1, ..., −1, 1, 2, ..., n } or of the set { −n, −n + 1, ..., n } such that π(i) = −π(−i) for all i. As a matrix group

Matrix group

In mathematics, a matrix group is a group G consisting of invertible matrices over some field K, usually fixed in advance, with operations of matrix multiplication and inversion. More generally, one can consider n × n matrices over a commutative ring R...

, it can be described as the group of n×n orthogonal matrices whose entries are all integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s. The representation theory of the hyperoctahedral group was described by according to .

In three dimensions, the hyperoctahedral group is known as O×S₂ where O≅S₄ is the octahedral group, and S₂ is a symmetric group (equivalently, cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry

Octahedral symmetry

150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

. In two dimensions, the hyperoctahedral group is known as the dihedral group of order eight acting naturally on a square or a diamond.

By dimension

Hyperoctahedral groups can be named as C_n, a bracket notation, or as a Coxeter group graph:

n	Symmetry group	Cn	Coxeter notation Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... graph	Symmetry order	Structure	Related regular and uniform polytope Polytope In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions... s
2	D4 (*44)	C2	[4]		222! = 8		Square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... , octagon
3	Oh Octahedral symmetry 150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...  (*432)	C3	[4,3]		233! = 48		Cube Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... , octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... , truncated cube Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices.... , truncated octahedron Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... , cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,... , rhombicuboctahedron Rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...  ...
4		C4	[4,3,3]		244! = 384		Tesseract Tesseract In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8... , 16-cell 16-cell In four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century.... , 24-cell, truncated tesseract Truncated tesseract In geometry, a truncated tesseract is a uniform polychoron formed as the truncation of the regular tesseract.There are three trunctions, including a bitruncation, and a tritruncation, which creates the truncated 16-cell.... , truncated 16-cell, rectified tesseract Rectified tesseract In geometry, the rectified tesseract, or rectified 8-cell is a uniform polychoron bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra.... , rectified 16-cell, cantellated tesseract Cantellated tesseract In four-dimensional geometry, a cantellated tesseract is a convex uniform polychoron, being a cantellation of the regular tesseract.There are four degrees of cantellations of the tesseract including with permutations truncations... , runcinated tesseract Runcinated tesseract In four-dimensional geometry, a runcinated tesseract is a convex uniform polychoron, being a runcination of the regular tesseract.... , ...
5		C5	[4,3,3,3]		255! = 3840		Penteract Penteract In five dimensional geometry, a 5-cube is a name for a five dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract hypercells.... , pentacross Pentacross In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell hypercells.... , truncated penteract, truncated pentacross, rectified penteract, rectified pentacross Rectified pentacross In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the... , ...
6		C6	[4,34]		266! = 46080
...
n		Cn	[4,3n-2]	...	2nn!

Subgroups

There is a notable index two subgroup, corresponding to the Coxeter group D_n and the symmetries of the demihypercube. Viewed as a wreath product, there are two natural maps from the hyperoctahedral group to the cyclic group of order 2: one map coming from "multiply the signs of all the elements" (in the n copies of ), and one map coming from the parity of the permutation. Multiplying these together yields a third map and the kernel of this map is the Coxeter group In terms of signed permutations, thought of as matrices, this third map is simply the determinant, while the first two correspond to "multiplying the non-zero entries" and "parity of the underlying (unsigned) permutation", which are not generally meaningful for matrices, but are in the case due to the coincidence with a wreath product.

The kernels of these three maps are all three index two subgroups of the hyperoctahedral group, as discussed in H₁: Abelianization below, and their intersection is the derived subgroup, of index 4 (quotient the Klein 4-group), which corresponds to the rotational symmetries of the demihypercube.

In the other direction, the center is the subgroup of scalar matrices, {±1}; geometrically, quotienting out by this corresponds to passing to the projective orthogonal group

Projective orthogonal group

In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = A quadratic space is a vector space V together with a quadratic form Q; the Q is dropped from notation when it is clear. on the associated projective...

.

In dimension 2 these groups completely describe the hyperoctahedral group, which is the dihedral group Dih₄ of order 8, and is an extension 2.V (of the 4-group by a cyclic group of order 2). In general, passing to the subquotient (derived subgroup, mod center) is the symmetry group of the projective demihypercube.

Homology

The group homology of the hyperoctahedral group is similar to that of the symmetric group, and exhibits stabilization, in the sense of stable homotopy theory

Stable homotopy theory

In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...

.

H₁: abelianization

The first homology group, which agrees with the abelianization, stabilizes at the Klein four-group

Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...

, and is given by:
This is easily seen directly: the elements are order 2 (which is non-empty for ), and all conjugate, as are the transpositions in (which is non-empty for ), and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, and either of these classes can be sent independently to as they are two separate classes. The maps are explicitly given as "the product of the signs of all the elements" (in the n copies of ), and the sign of the permutation. Multiplying these together yields a third non-trivial map (the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

 of the matrix, which sends both these classes to ), and together with the trivial map these form the 4-group.

H₂: Schur multipliers

The second homology groups, known classically as the Schur multiplier

Schur multiplier

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...

s, were computed in .

They are: