Signed symmetric group

This is a variation of symmetric group|Find other variations of symmetric group |

Definition

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

1. It is the 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 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 generalized symmetric group $S(2,n)$.
7. It is a 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

Function	Value	Explanation
order	$2^n \cdot n!$
exponent	lcm of $1,2,\dots,n$ (?)

Particular cases

Value of $n$	Order = $2^n \cdot n!$	GAP ID	Common name
0	1	(1,1)	trivial group
1	2	(2,1)	cyclic group:Z2
2	8	(8,3)	dihedral group:D8
3	48	(48,48)	direct product of S4 and Z2
4	384	(384,5602)

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))