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Point Group Tables of Oh(m-3m)

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Character Table of the group Oh(m-3m)*
Oh(m-3m)#14210032110-1-4m100-3m110functions
Mult.-1638616386·
A1gΓ1+1111111111x2+y2+z2
A1uΓ1-11111-1-1-1-1-1·
A2gΓ2+1-111-11-111-1·
A2uΓ2-1-111-1-11-1-11·
EgΓ3+202-10202-10(2z2-x2-y2,x2-y2)
EuΓ3-202-10-20-210·
T2uΓ5-3-1-101-3110-1·
T2gΓ5+3-1-1013-1-101(xy,xz,yz)
T1uΓ4-31-10-1-3-1101(x,y,z)
T1gΓ4+31-10-131-10-1(Jx,Jy,Jz)



Subgroups of the group Oh(m-3m)
SubgroupOrderIndex
Oh(m-3m)481
Td(-43m)242
O(432)242
Th(m-3)242
T(23)124
D3d(-3m)124
C3v(3m)68
D3(32)68
C3i(-3)68
C3(3)316
D4h(4/mmm)163
D2d(-42m)86
C4v(4mm)86
D4(422)86
C4h(4/m)86
S4(-4)412
C4(4)412
D2h(mmm)86
C2v(mm2)412
D2(222)412
C2h(2/m)412
C2(2)224
Cs(m)224
Ci(-1)224
C1(1)148

[ Subduction tables ]

Multiplication Table of irreducible representations of the group Oh(m-3m)
Oh(m-3m)A1gA1uA2gA2uEuEgT2uT2gT1uT1g
A1gA1gA1uA2gA2uEuEgT2uT2gT1uT1g
A1u·A1gA2uA2gEgEuT2gT2uT1gT1u
A2g··A1gA1uEuEgT1uT1gT2uT2g
A2u···A1gEgEuT1gT1uT2gT2u
Eu····A1g+A2g+EgA1u+A2u+EuT2g+T1gT2u+T1uT2g+T1gT2u+T1u
Eg·····A1g+A2g+EgT2u+T1uT2g+T1gT2u+T1uT2g+T1g
T2u······A1g+Eg+T2g+T1gA1u+Eu+T2u+T1uA2g+Eg+T2g+T1gA2u+Eu+T2u+T1u
T2g·······A1g+Eg+T2g+T1gA2u+Eu+T2u+T1uA2g+Eg+T2g+T1g
T1u········A1g+Eg+T2g+T1gA1u+Eu+T2u+T1u
T1g·········A1g+Eg+T2g+T1g

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
Oh(m-3m)A1gA1uA2gA2uEuEgT2uT2gT1uT1g
[A1g x A1g]1·········
[A1u x A1u]1·········
[A2g x A2g]1·········
[A2u x A2u]1·········
[Eu x Eu]1····1····
[Eg x Eg]1····1····
[T2u x T2u]1····1·1··
[T2g x T2g]1····1·1··
[T1u x T1u]1····1·1··
[T1g x T1g]1····1·1··


Antisymmetrized Products of Irreps
Oh(m-3m)A1gA1uA2gA2uEuEgT2uT2gT1uT1g
{A1g x A1g}··········
{A1u x A1u}··········
{A2g x A2g}··········
{A2u x A2u}··········
{Eu x Eu}··1·······
{Eg x Eg}··1·······
{T2u x T2u}·········1
{T2g x T2g}·········1
{T1u x T1u}·········1
{T1g x T1g}·········1


Irreps Decompositions
Oh(m-3m)A1gA1uA2gA2uEuEgT2uT2gT1uT1g
V········1·
[V2]1····1·1··
[V3]···1··1·2·
[V4]2····2·2·1
A·········1
[A2]1····1·1··
[A3]··1····1·2
[A4]2····2·2·1
[V2]xV···11·2·3·
[[V2]2]3····3·3·1
{V2}·········1
{A2}·········1
{[V2]2}··1··1·2·2

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1gA1uA2gA2uEuEgT2uT2gT1uT1g
A1g········x·
A1u·········x
A2g······x···
A2u·······x··
Eu·······x·x
Eg······x·x·
T2u··x··x·x·x
T2g···xx·x·x·
T1ux····x·x·x
T1g·x··x·x·x·

[ Note: x means allowed ]


Raman Selection Rules
RamanA1gA1uA2gA2uEuEgT2uT2gT1uT1g
A1gx····x·x··
A1u·x··x·x···
A2g··x··x···x
A2u···xx···x·
Eu·x·xx·x·x·
Egx·x··x·x·x
T2u·x··x·x·x·
T2gx····x·x·x
T1u···xx·x·x·
T1g··x··x·x·x

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group Oh(m-3m)
L2L+1A1gA1uA2gA2uEuEgT2uT2gT1uT1g
011·········
13········1·
25·····1·1··
37···1··1·1·
491····1·1·1
511····1·1·2·
6131·1··1·2·1
715···11·2·2·
8171····2·2·2
919·1·11·2·3·
10211·1··2·3·2


* 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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