Dihedral Group (Subscript[D, 6])

Author

Eric W. Weisstein
October 6, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/DihedralGroupD6.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DihedralGroupD6.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Groups`

Representation

letters=Prepend[CharacterRange["A","K"],"1"];

Permutation Group

pg=DihedralGroup[6]

prules=Thread[pg->letters];
nrules=Thread[pg->Range[Length[pg]]];

GroupQ[pg]

Properties

Abelian

AbelianGroupQ[pg]

Cycle Index

DihedralGroupIndex[6,x]/.x[n_]:>Subscript[x,n]//TraditionalForm

Conjugacy Classes

ConjugacyClasses[pg]/.prules

Length/@%

Subgroups

Length[subg=Subgroups[pg]]//Timing

subg/.prules

Length/@subg

Select[subg,NormalSubgroupQ[#,pg]&]/.prules

Multiplication Table

GroupMultiplicationTable[pg,prules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[pg,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[(Subscript[D, 6])],
DisplayFunction->Identity]&/@{Hue[#]&,GrayLevel}]]

Powers

Table[GroupTimes[Sequence@@Table[#,{n}]]&/@pg,{n,Length[pg]}]/.prules//TableForm

Count[#,"1"]&/@(Table[GroupTimes[Sequence@@Table[#,{n}]]&/@pg,{n,Length[pg]}]/.prules)

Cycle Graph

MaximalGraphCycles[DihedralGroup[6]]

ShowLabeledGraph[RadialEmbedding[CycleGraph[DihedralGroup[6]]]]