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.

The DihedralGroup( n ) command returns a dihedral group, either as a permutation group or a group defined by generators and defining relations. By default, a permutation group is returned, but a finitely presented group can be requested by passing the option 'form' = "fpgroup".

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.

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

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

${\mathbf{D}}_{13}$

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

${\mathbf{D}}_{13}$

$\mathrm{DihedralGroup}\left(13\,\mathrm{form}=''permgroup''\right)$

${\mathbf{D}}_{13}$

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

$6k$

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

$\mathrm{false}$

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

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