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

Author

Eric W. Weisstein
September 26, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/FiniteGroupC2C2C2.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FiniteGroupC2C2C2.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Representation

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

Modulo Multiplication Group

mmg=ModuloMultiplicationGroup[24]

GroupQ[mmg]

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

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

ModuloMultiplicationPhi/@{24}

Properties

AbelianGroupQ[mmg]

Multiplication Table

GroupMultiplicationTable[mmg,mrules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[mmg,nrules][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[HoldForm[(Subscript[C, 2])\[Cross](Subscript[C, 2])\[Cross](Subscript[C, 2])]],
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

conj=ConjugacyClasses[mmg]/.rules

Length/@conj

Length[%]

Cycle Graph

GroupCycles[mmg]/.rules

ShowLabeledGraph[CycleGraph[mmg]]

Subgroups

(subg=Subgroups[mmg])/.rules

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

Length[%]

Select[Subgroups[mmg],NormalSubgroupQ[#,mg]&]/.rules