Tetrahedral Group

Author

Eric W. Weisstein
January 15, 2019

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

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

©2019 Wolfram Research, Inc. except for portions noted otherwise

(Subscript[T, d])

FiniteGroupData["Td"]

FiniteGroupData["Tetrahedral","AlternateStandardNames"]

bug(365915)

FiniteGroupData[{"CrystallographicPointGroup","Td"}]

FiniteGroupData[{"CrystallographicPointGroup","Td"},"IsomorphicGroups"]

FiniteGroupData["Tetrahedral","Order"]

FiniteGroupData["Tetrahedral","Notation"]

FiniteGroupData["Tetrahedral","Classes"]

FiniteGroupData["Tetrahedral","ConjugacyClasses"]

FiniteGroupData["Tetrahedral","ConjugacyClassSizes"]

ArrayPlot[FiniteGroupData["Tetrahedral","MultiplicationTable"],ColorFunction->#]&/@{Hue,GrayLevel}

FiniteGroupData["Tetrahedral","CayleyGraph"]

UndirectedGraph[%]//ToEntity

spacerep=FiniteGroupData["Tetrahedral","SpaceRepresentation"]

matrixrep=FiniteGroupData["Tetrahedral","MatrixRepresentation"]

spacerep==matrixrep

FiniteGroupData["Tetrahedral","ElementNames"]

Dimensions[mt=FiniteGroupData["Tetrahedral","MultiplicationTable"]]

mt==With[{m=matrixrep},Outer[Dot,m,m,1]/.Thread[m->Range[Length[m]]]]

mt==With[{m=permutationrep},Outer[PermutationReplace,m,m,1]/.Thread[m->Range[Length[m]]]]

T

Select[FiniteGroupData[All],MemberQ[FiniteGroupData[#,"AlternateStandardNames"],"T"]&]

FiniteGroupData["T"]

FiniteGroupData[{"CrystallographicPointGroup","T"}]

FiniteGroupData[{"CrystallographicPointGroup",28},"IsomorphicGroups"]

FiniteGroupData[{"CrystallographicPointGroup",28},"AlternateStandardNames"]

FiniteGroupData["A4"]

bug(365915)

FiniteGroupData[{"CrystallographicPointGroup",28},"Notation"]

FiniteGroupData[{"CrystallographicPointGroup",28},"ElementNames"]

FiniteGroupData[{"CrystallographicPointGroup",28},"Classes"]

FiniteGroupData[{"CrystallographicPointGroup",28},"ConjugacyClasses"]

FiniteGroupData[{"CrystallographicPointGroup",28},"ConjugacyClassSizes"]

ArrayPlot[FiniteGroupData[{"CrystallographicPointGroup",28},"MultiplicationTable"],ColorFunction->#]&/@{Hue,GrayLevel}

FiniteGroupData[{"CrystallographicPointGroup",28},"CayleyGraph"]

UndirectedGraph[%]//ToEntity

matrixrep=FiniteGroupData[{"CrystallographicPointGroup",28},"MatrixRepresentation"]

spacerep=FiniteGroupData[{"CrystallographicPointGroup",28},"SpaceRepresentation"]

permutationrep=FiniteGroupData[{"CrystallographicPointGroup",28},"PermutationRepresentation"]

FiniteGroupData[{"CrystallographicPointGroup",28},"ElementNames"]

Dimensions[mt=FiniteGroupData[{"CrystallographicPointGroup",28},"MultiplicationTable"]]

mt==With[{m=matrixrep},Outer[Dot,m,m,1]/.Thread[m->Range[Length[m]]]]

mt==With[{m=permutationrep},Dimensions[Outer[PermutationReplace,m,m,1]/.Thread[m->Range[Length[m]]]]]

(Subscript[T, h])

FiniteGroupData["Th"]

FiniteGroupData[{"CrystallographicPointGroup","Th"}]

FiniteGroupData[{"CrystallographicPointGroup","Th"},"IsomorphicGroups"]

FiniteGroupData[{"CrystallographicPointGroup",29},"AlternateStandardNames"]

FiniteGroupData[{"CrystallographicPointGroup",29},"Notation"]

FiniteGroupData[{"CrystallographicPointGroup",28},"ElementNames"]

FiniteGroupData[{"CrystallographicPointGroup",29},"Classes"]

FiniteGroupData[{"CrystallographicPointGroup",29},"ConjugacyClasses"]

FiniteGroupData[{"CrystallographicPointGroup",29},"ConjugacyClassSizes"]

ArrayPlot[FiniteGroupData[{"CrystallographicPointGroup",29},"MultiplicationTable"],ColorFunction->#]&/@{Hue,GrayLevel}

FiniteGroupData[{"CrystallographicPointGroup",29},"CayleyGraph"]

UndirectedGraph[%]//ToEntity

matrixrep=FiniteGroupData[{"CrystallographicPointGroup",29},"MatrixRepresentation"]

spacerep=FiniteGroupData[{"CrystallographicPointGroup",29},"SpaceRepresentation"]

matrixrep==spacerep

permutationrep=FiniteGroupData[{"CrystallographicPointGroup",29},"PermutationRepresentation"]

FiniteGroupData[{"CrystallographicPointGroup",28},"ElementNames"]

Dimensions[mt=FiniteGroupData[{"CrystallographicPointGroup",29},"MultiplicationTable"]]

mt==With[{m=matrixrep},Outer[Dot,m,m,1]/.Thread[m->Range[Length[m]]]]

mt==With[{m=spacerep},Outer[Dot,m,m,1]/.Thread[m->Range[Length[m]]]]

mt==With[{m=permutationrep},Outer[PermutationReplace,m,m,1]/.Thread[m->Range[Length[m]]]]

(Subscript[T, d]) (pre-FiniteGroupData)

tdrules=Thread[PointGroup["Td"]->Range[24]];

Group

GroupQ[PointGroup["Td"]]//Timing

Abelian

AbelianGroupQ[PointGroup["Td"]]

Conjugacy Classes

ConjugacyClasses[PointGroup["Td"]]/.rules

Length/@%

Subgroups

SubgroupQ[Sequence@@PointGroup/@{"T","Td"}]//Timing

(subg=Subgroups[PointGroup["Td"]]/.rules)//Timing

Length/@subg

Multiplication Table

GroupMultiplicationTable[PointGroup["Td"],tdrules]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[PointGroup["Td"],tdrules][[1]],ColorFunction->#,Frame->False,Mesh->True,PlotLabel->TraditionalForm[(Subscript[T, d])],
DisplayFunction->Identity]&/@{Hue,GrayLevel}]]

Cycle Graph

MaximalGraphCycles[PointGroup["Td"]]

ShowLabeledGraph[RadialEmbedding[CycleGraph[PointGroup["Td"]]]]

T (pre-FiniteGroupData)

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

Permutation Group

pg=AlternatingGroup[4];

GroupQ[pg]

prules=Thread[pg->letters];

Matrix Group

mg=PointGroup["T"];

GroupQ[mg]

mrules=Thread[mg->letters];

Abelian

AbelianGroupQ/@{pg,mg}

Conjugacy Classes

{pconj,mconj}={ConjugacyClasses[pg]/.prules,ConjugacyClasses[mg]/.mrules}

{Length/@pconj,Length/@mconj}

Subgroups

{(psubg=Subgroups[pg])/.prules,(msubg=Subgroups[mg])/.mrules}

Length/@{psubg,msubg}

{Length/@psubg,Length/@msubg}

{Select[msubg,NormalSubgroupQ[#,mg]&]/.mrules,
Select[psubg,NormalSubgroupQ[#,pg]&]/.prules}

Multiplication Table

GroupMultiplicationTable[pg,prules]//TraditionalForm

GroupMultiplicationTable[mg,mrules]//TraditionalForm

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[pg,Thread[pg->Range[12]]][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[T],
DisplayFunction->Identity]&/@{Hue,GrayLevel}]]

Show[GraphicsArray[ListDensityPlot[Reverse@GroupMultiplicationTable[mg,Thread[mg->Range[12]]][[1]],ColorFunction->#,Frame->False,Mesh->True,
PlotLabel->TraditionalForm[T],
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)

Polyhedra

Vector[TruncatedTetrahedron]={1,1,3};
Vector[Tetrahedron]={1,1,1};
Vector[Icosahedron]={1,0,GoldenRatio};
Vector[Octahedron]={1,0,0};
Vector[Cuboctahedron]={1,0,1};

polys=#[[1,1,1]]&/@DownValues[Vector]

<<ConvexHull`

Show[GraphicsArray[Partition[
Graphics3D[#,Boxed->False]&/@Function[v,ConvexHull3D[Union[Chop/@Simplify[#.v&/@PointGroup["T"]]]]]/@(N/@Vector/@polys),
5,5,{1,1},{}]
]]//Timing

Cycle Graph

{MaximalGroupCycles[pg]/.prules,MaximalGroupCycles[mg]/.mrules}

ShowLabeledGraph[RadialEmbedding[CycleGraph[mg]]]