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Point Group Tables of O(432)

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Character Table of the group O(432)*
O(432)#14210032110functions
Mult.-16386·
A1Γ111111x2+y2+z2
A2Γ21-111-1·
EΓ3202-10(2z2-x2-y2,x2-y2)
T2Γ53-1-101(xy,xz,yz)
T1Γ431-10-1(x,y,z),(Jx,Jy,Jz)



Subgroups of the group O(432)
SubgroupOrderIndex
O(432)241
T(23)122
D3(32)64
D4(422)83
C4(4)46
C3(3)38
D2(222)46
C2(2)212
C1(1)124

[ Subduction tables ]

Multiplication Table of irreducible representations of the group O(432)
O(432)A1A2ET2T1
A1A1A2ET2T1
A2·A1ET1T2
E··A1+A2+ET2+T1T2+T1
T2···A1+E+T2+T1A2+E+T2+T1
T1····A1+E+T2+T1

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
O(432)A1A2ET2T1
[A1 x A1]1····
[A2 x A2]1····
[E x E]1·1··
[T2 x T2]1·11·
[T1 x T1]1·11·


Antisymmetrized Products of Irreps
O(432)A1A2ET2T1
{A1 x A1}·····
{A2 x A2}·····
{E x E}·1···
{T2 x T2}····1
{T1 x T1}····1


Irreps Decompositions
O(432)A1A2ET2T1
V····1
[V2]1·11·
[V3]·1·12
[V4]2·221
A····1
[A2]1·11·
[A3]·1·12
[A4]2·221
[V2]xV·1123
[[V2]2]3·331
{V2}····1
{A2}····1
{[V2]2}·1122

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1A2ET2T1
A1····x
A2···x·
E···xx
T2·xxxx
T1x·xxx

[ Note: x means allowed ]


Raman Selection Rules
RamanA1A2ET2T1
A1x·xx·
A2·xx·x
Exxxxx
T2x·xxx
T1·xxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group O(432)
L2L+1A1A2ET2T1
011····
13····1
25··11·
37·1·11
491·111
511··112
61311121
715·1122
8171·222
91911123
102111232


* C. J. Bradley and A. P. Cracknell (1972) The Mathematical Theory of Symmetry in Solids Clarendon Press - Oxford
* Simon L. Altmann and Peter Herzig (1994). Point-Group Theory Tables. Oxford Science Publications.

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