GroupTheory
DihedralGroup
construct a dihedral group of a given degree
Calling Sequence
Parameters
Description
Examples
Compatibility
DihedralGroup( n )
DihedralGroup( n, s )
n
-
: algebraic : an expression understood to be a positive integer or ∞
s
: equation : (optional) equation of the form form = "fpgroup" or form = "permgroup" (default)
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=∞, 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=∞ 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.
withGroupTheory:
G≔DihedralGroup13
G≔D13
GroupOrderG
26
G≔DihedralGroup13,form=fpgroup
G≔DihedralGroup17,form=permgroup
G≔D17
34
AreIsomorphicDihedralGroup3,Symm3
true
GroupOrderDihedralGroup3k
6k
IsNilpotentDihedralGroup6kassumingk::posint
false
IsNilpotentDihedralGroup2a4bassumingposint
IsFrobeniusGroupDihedralGroup7
IsFrobeniusGroupDihedralGroup6
DrawCayleyTableDihedralGroup5,conjugacy=true
ClassNumberDihedralGroup6nassumingn::posint
3n+3
ExponentDihedralGroup2n+1assumingn::posint
4n+2
IsPerfectOrderClassesGroupDihedralGroup9
IsPerfectOrderClassesGroupDihedralGroup10
G≔DihedralGroup∞
G≔D∞
IsNilpotentG
IsSupersolubleG
IdentifyFrobeniusGroupDihedralGroup11
22,1
DisplayCharacterTableDihedralGroup5
C
1a
2a
5a
5b
|C|
1
5
2
chi__1
chi__2
−1
chi__3
0
−125−−135
−135−−125−1
chi__4
caygr≔CayleyGraphDihedralGroup4
caygr≔Graph 1: a directed graph with 8 vertices and 16 arcs
GraphTheory:-DrawGraphcaygr
The GroupTheory[DihedralGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[DicyclicGroup]
GroupTheory[GroupOrder]
GroupTheory[IsNilpotent]
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