The dihedral group of degree $n$ is the symmetry group of an $n$-sided regular polygon for $n\>2$. It is generated by a reflection (of order $2$), and a rotation (of order $n$). It acts as a permutation group on the vertices of the regular $n$-sided polygon.

For $n=1$, the dihedral group is a cyclic group of order $2$.  For $n=2$, the dihedral group is the non-cyclic group of order $4$, also known as the Klein $4$-group.

If $n=\mathrm{\infty }$, then an infinite dihedral group (a free product of two groups of order two, or the holomorph of an infinite cyclic group) is returned as a finitely presented group.

The DihedralGroup( n ) command returns a dihedral group, either as a permutation group or a group defined by generators and defining relations. By default, if n is a positive integer, then a permutation group is returned, but a finitely presented group can be requested by passing the option 'form' = "fpgroup". If $n=\mathrm{\infty }$ then a finitely presented group is returned, regardless of any form option passed.

If the value of the parameter n is not numeric, then a symbolic group representing the dihedral group of the indicated degree is returned.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{DihedralGroup}\left(13\right)$

$G≔{\mathbf{D}}_{13}$

$\mathrm{GroupOrder}\left(G\right)$

$26$

$G≔\mathrm{DihedralGroup}\left(13\,\mathrm{form}=''fpgroup''\right)$

$G≔{\mathbf{D}}_{13}$

$G≔\mathrm{DihedralGroup}\left(17\,\mathrm{form}=''permgroup''\right)$

$G≔{\mathbf{D}}_{17}$

$\mathrm{GroupOrder}\left(G\right)$

$34$

$\mathrm{AreIsomorphic}\left(\mathrm{DihedralGroup}\left(3\right)\,\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{DihedralGroup}\left(3k\right)\right)$

$6k$

$\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(6k\right)\right)\mathbf{assuming}k::\mathrm{posint}$

$\mathrm{false}$

$\mathrm{IsNilpotent}\left(\mathrm{DihedralGroup}\left(2^a{4}^b\right)\right)\mathbf{assuming}\mathrm{posint}$

$\mathrm{true}$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(7\right)\right)$

$\mathrm{true}$

$\mathrm{IsFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$

$\mathrm{false}$

$\mathrm{DrawCayleyTable}\left(\mathrm{DihedralGroup}\left(5\right)\,'\mathrm{conjugacy}'=\mathrm{true}\right)$

$\mathrm{ClassNumber}\left(\mathrm{DihedralGroup}\left(6n\right)\right)\mathbf{assuming}n::\mathrm{posint}$

$3n+3$

$\mathrm{Exponent}\left(\mathrm{DihedralGroup}\left(2n+1\right)\right)\mathbf{assuming}n::\mathrm{posint}$

$4n+2$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(9\right)\right)$

$\mathrm{true}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{DihedralGroup}\left(10\right)\right)$

$\mathrm{false}$

$G≔\mathrm{DihedralGroup}\left(\mathrm{\infty }\right)$

$G≔{\mathbf{D}}_{\mathrm{\infty }}$

$\mathrm{IsNilpotent}\left(G\right)$

$\mathrm{false}$

$\mathrm{IsSupersoluble}\left(G\right)$

$\mathrm{true}$

$\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{DihedralGroup}\left(11\right)\right)$

$22,1$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{DihedralGroup}\left(5\right)\right)\right)$

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    JEYlJShfc3lzbGliRzYnLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiM0Y8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIyRjxGRUZIRkstRjc2I1EiMUY8'>LUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYnLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiM0Y8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIyRjxGRUZIRkstRjc2I1EiMUY8</Equation></Text-field></Input></Group></Table-Cell><Table-Cell columnspan='1' backgroundstyle='0' rowspan='1' fillcolor='[255,255,255]' visible='true'><Group view='presentation' hide-input='false' hide-output='false' inline-output='false' drawlabel='true'><Input><Text-field alignment='centred' style='Text' layout='Normal'><Equation executable='false\

    ' style='2D Input' input-equation='' display='LUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYlLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiMkY8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIzRjxGRUZI'>LUklbXJvd0c2JCUqcHJvdGVjdGVkRy8lK21vZHVsZW5hbWVHSSxUeXBlc2V0dGluZ0c2JEYlJShfc3lzbGliRzYlLUklbXN1cEc2JEYlL0YnRig2JC1JKG1mZW5jZWRHNiRGJS9GJ0YoNiMtSSNtbkc2JEYlL0YnRig2I1ErJnVtaW51czA7MTYiLUkmbWZyYWNHNiRGJS9GJ0YoNiUtRjc2I1EiMkY8LUY3NiNRIjVGPC8lKWJldmVsbGVkR1EldHJ1ZUY8LUkjbW9HNiRGJS9GJ0YoNiNRKCZtaW51cztGPC1GLTYkRjEtRj42JS1GNzYjUSIzRjxGRUZI</Equation></Text-field></Input></Group></Table-Cell></Table-Row></Table></Worksheet>", state = "", minimal = true

$\mathrm{caygr}≔\mathrm{CayleyGraph}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{caygr}≔\mathrm{Graph\; 1:\; a\; directed\; graph\; with\; 8\; vertices\; and\; 16\; arc(s)}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{caygr}\right)$

The GroupTheory[DihedralGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.