Signed symmetric group

Definition
The signed symmetric group or hyperoctahedral group of degree $$n$$ is defined in the following equivalent ways:


 * 1) It is the defining ingredient::external wreath product of the cyclic group of order two $$\mathbb{Z}_2$$ with the symmetric group of degree $$n$$ with its natural action on a set of size $$n$$. In symbols, it is $$\mathbb{Z}_2 \wr S_n$$.
 * 2) It is the defining ingredient::external semidirect product of an elementary abelian group of order $$2^n$$ and a symmetric group of degree $$n$$, acting as coordinate permutations in the natural way.
 * 3) It is the subgroup of the general linear group over integers $$GL(n,\mathbb{Z})$$ comprising all matrices which have exactly one nonzero entry in each row and exactly one nonzero entry in each column, and the nonzero entries are all $$\pm 1$$. More generally, it can be realized using these matrices over any ring of characteristic not equal to two.
 * 4) It is the group of symmetries of the hyperoctahedron in $$\mathbb{R}^n$$.
 * 5) It is the centralizer in the symmetric group of degree $$2n$$ of a permutation that is a product of $$n$$ disjoint transpositions.
 * 6) It is the defining ingredient::generalized symmetric group $$S(2,n)$$.
 * 7) It is a defining ingredient::Coxeter group with a particular kind of Coxeter presentation (for more, see signed symmetric group of finite degree is a Coxeter group)

Arithmetic functions
All these function values are

GAP implementation
The groups can be constructed using GAP, with the help of the functions WreathProduct, CyclicGroup, and SymmetricGroup. For a given $$n$$, the signed symmetric group is given by:

WreathProduct(CyclicGroup(2),SymmetricGroup(n))

For instance, for $$n = 4$$, the group can be defined as:

WreathProduct(CyclicGroup(2),SymmetricGroup(4))