# DihedralGroup

represents the dihedral group of order 2n.

# Details

• The degree n of must be a positive integer.
• is isomorphic to CyclicGroup[2] and represented by default as a permutation group on the points {1,2}.
• is isomorphic to AbelianGroup[{2,2}] and represented by default as a permutation group on the points {1,2,3,4}.
• For n3, is represented by default as a permutation group on the points {1,,n}.

# Background & Context

• represents the dihedral group of order (also denoted or ) for a given positive integer n. For , the default representation of is as a permutation group on the symbols . The special cases and are isomorphic to CyclicGroup[2] and AbelianGroup[{2,2}], respectively, and are represented by default as a permutation group on the symbols and , respectively.
• Mathematically, the dihedral group consists of the symmetries of a regular -gon, namely its rotational symmetries and reflection symmetries. In particular, consists of elements (rotations) and (reflections), which combine to transform under its group operation according to the identities , , and , where addition and subtraction are performed modulo . is a permutation group, but for , the operations of reflection and rotation fail to commute in general, meaning is nonabelian for .
• Dihedral groups are important in the analysis of regular structures, including in the determination of properties for symmetric chemical compounds and in crystallography.
• The usual group theoretic functions may be applied to DiherdalGroup[n], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the dihedral group are available via FiniteGroupData[{"DihedralGroup",n},"prop"].
• DihedralGroup is related to a number of other symbols. is isomorphic to the semidirect product of CyclicGroup[n] and CyclicGroup[2] (with the latter acting on the former by inversion), and for even, is isomorphic to the direct product of DihedralGroup[n/2] and CyclicGroup[2]. For , the dihedral group is a subgroup of the symmetric group . Other infinite families of finite groups built into the Wolfram Language that are parametrized by integers include AbelianGroup, AlternatingGroup, CyclicGroup and SymmetricGroup.

# Examples

open allclose all

## Basic Examples(3)

Number of elements of a dihedral group:

Permutation generators of a dihedral group:

Elements of a permutation representation of a dihedral group:

## Scope(1)

for any positive integer n:

## Properties & Relations(1)

and are the only dihedral commutative groups:

## Possible Issues(1)

and are special as permutation groups because they are not subgroups of and , respectively. Their permutation representations require larger supports:

Wolfram Research (2010), DihedralGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/DihedralGroup.html.

#### Text

Wolfram Research (2010), DihedralGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/DihedralGroup.html.

#### CMS

Wolfram Language. 2010. "DihedralGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DihedralGroup.html.

#### APA

Wolfram Language. (2010). DihedralGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DihedralGroup.html

#### BibTeX

@misc{reference.wolfram_2021_dihedralgroup, author="Wolfram Research", title="{DihedralGroup}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/DihedralGroup.html}", note=[Accessed: 25-June-2022 ]}

#### BibLaTeX

@online{reference.wolfram_2021_dihedralgroup, organization={Wolfram Research}, title={DihedralGroup}, year={2010}, url={https://reference.wolfram.com/language/ref/DihedralGroup.html}, note=[Accessed: 25-June-2022 ]}