Finite Group (Subscript[C, 2])×(Subscript[C, 6])

Author

Eric W. Weisstein
October 6, 2003

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/FiniteGroupC2C6.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FiniteGroupC2C6.html.

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

Initialization

<<Groups`

Representation

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

Modulo Multiplication Group

mmg=ModuloMultiplicationGroup[21]

GroupQ[mmg]

rules=Thread[mmg->letters];
nrules=Thread[mmg->Range[Length[mmg]]];

Position[CharacteristicFactors[ModuloMultiplicationPhi[#]]&/@Range[10000],{2,6}]//Flatten//Timing

ModuloMultiplicationPhi/@{21,28,36,42}

Properties

AbelianGroupQ[mmg]

Multiplication Table

GroupMultiplicationTable[mmg,rules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[mmg,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[HoldForm[(Subscript[C, 2])\[Cross](Subscript[C, 6])]],
DisplayFunction->Identity]&/@{Hue[#]&,GrayLevel}]]

Powers

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

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

Conjugacy Classes

cong=ConjugacyClasses[mmg]/.rules

Length/@cong

Length[%]

Subgroups

Timing[(subg=Subgroups[mmg])/.rules]

Through[{Length,First}[#]]&/@Split[Length/@subg]

Length[%]

Select[subg,NormalSubgroupQ[#,mmg]&]/.rules

Length[%75]

Cycle Graph

MaximalGraphCycles[mmg]/.rules

ShowLabeledGraph[RadialEmbedding[CycleGraph[mmg]]]