Signed symmetric group
This is a variation of symmetric group|Find other variations of symmetric group |
The signed symmetric group or hyperoctahedral group of degree is defined in the following equivalent ways:
- It is the external wreath product of the cyclic group of order two with the symmetric group of degree with its natural action on a set of size . In symbols, it is .
- It is the external semidirect product of an elementary abelian group of order and a symmetric group of degree , acting as coordinate permutations in the natural way.
- It is the subgroup of the general linear group over integers 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 . More generally, it can be realized using these matrices over any ring of characteristic not equal to two.
- It is the group of symmetries of the hyperoctahedron in .
- It is the centralizer in the symmetric group of degree of a permutation that is a product of disjoint transpositions.
- It is the generalized symmetric group .
- 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)
All these function values are
|exponent||lcm of (?)|
|Value of||Order =||GAP ID||Common name|
|3||48||(48,48)||direct product of S4 and Z2|
For instance, for , the group can be defined as: