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Point Group Tables of D4(422)

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Character Table of the group D4(422)*
D4(422)#124210021-10functions
Mult.-11222·
A1Γ111111x2+y2,z2
A2Γ2111-1-1z,Jz
B1Γ311-11-1x2-y2
B2Γ411-1-11xy
EΓ52-2000(x,y),(xz,yz),(Jx,Jy)



Subgroups of the group D4(422)
SubgroupOrderIndex
D4(422)81
C4(4)42
D2(222)42
C2(2)24
C1(1)18

[ Subduction tables ]

Multiplication Table of irreducible representations of the group D4(422)
D4(422)A1A2B1B2E
A1A1A2B1B2E
A2·A1B2B1E
B1··A1A2E
B2···A1E
E····A1+A2+B1+B2

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D4(422)A1A2B1B2E
[A1 x A1]1····
[A2 x A2]1····
[B1 x B1]1····
[B2 x B2]1····
[E x E]1·11·


Antisymmetrized Products of Irreps
D4(422)A1A2B1B2E
{A1 x A1}·····
{A2 x A2}·····
{B1 x B1}·····
{B2 x B2}·····
{E x E}·1···


Irreps Decompositions
D4(422)A1A2B1B2E
V·1··1
[V2]2·111
[V3]·2113
[V4]41223
A·1··1
[A2]2·111
[A3]·2113
[A4]41223
[V2]xV13225
[[V2]2]61334
{V2}·1··1
{A2}·1··1
{[V2]2}12224

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1A2B1B2E
A1·x··x
A2x···x
B1···xx
B2··x·x
Exxxxx

[ Note: x means allowed ]


Raman Selection Rules
RamanA1A2B1B2E
A1x·xxx
A2·xxxx
B1xxx·x
B2xx·xx
Exxxxx

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group D4(422)
L2L+1A1A2B1B2E
011····
13·1··1
251·111
37·1112
4921112
51112113
61321223
71512224
81732224
91923225
102132335


* 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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