Hyperoctahedral groups can be named as Bn, a bracket notation, or as a Coxeter group graph:
Subgroups
There is a notable index two subgroup, corresponding to the Coxeter group Dn 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", and one map coming from the parity of the permutation. Multiplying these together yields a third map. The kernel of the first 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 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 below, and their intersection is the derived subgroup, of index 4, which corresponds to the rotational symmetries of the demihypercube. In the other direction, the center is the subgroup of scalar matrices, ; geometrically, quotienting out by this corresponds to passing to the projective orthogonal group. 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. In general, passing to the subquotient is the symmetry group of the projective demihypercube. in three dimensions, order 24 The hyperoctahedral subgroup, Dn by dimension: in three dimensions, order 24 in three dimensions, order 24 The chiral hyper-octahedral symmetry, is the direct subgroup, index 2 of hyper-octahedral symmetry. Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension: These groups have n orthogonal mirrors in n-dimensions.
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.
The first homology group, which agrees with the abelianization, stabilizes at the Klein four-group, and is given by: This is easily seen directly: the elements are order 2, and all conjugate, as are the transpositions in , 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", and the sign of the permutation. Multiplying these together yields a third non-trivial map, and together with the trivial map these form the 4-group.
