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Point Group Tables of D6h(6/mmm)

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Character Table of the group D6h(6/mmm)*
D6h(6/mmm)#1632z21202100-1-6-3mzm120m100functions
Mult.-122133122133·
A1gΓ1+111111111111x2+y2,z2
A1uΓ1-111111-1-1-1-1-1-1·
A2gΓ2+1111-1-11111-1-1Jz
A2uΓ2-1111-1-1-1-1-1-111z
B1gΓ3+1-11-11-11-11-11-1·
B1uΓ3-1-11-11-1-11-11-11·
B2gΓ4+1-11-1-111-11-1-11·
B2uΓ4-1-11-1-11-11-111-1·
E2uΓ6-2-1-1200-211-200·
E2gΓ6+2-1-12002-1-1200(x2-y2,xy)
E1uΓ5-21-1-200-2-11200(x,y)
E1gΓ5+21-1-20021-1-200(xz,yz),(Jx,Jy)
[Note: there are differences between Bradley & Cracknell (1972) and Altmann & Herzig (1994)]



Subgroups of the group D6h(6/mmm)
SubgroupOrderIndex
D6h(6/mmm)241
D3h(-62m)122
C6v(6mm)122
D6(622)122
C6h(6/m)122
D3d(-3m)122
C3h(-6)64
C6(6)64
C3v(3m)64
D3(32)64
C3i(-3)64
C3(3)38
D2h(mmm)83
C2v(mm2)46
D2(222)46
C2h(2/m)46
C2(2)212
Cs(m)212
Ci(-1)212
C1(1)124

[ Subduction tables ]

Multiplication Table of irreducible representations of the group D6h(6/mmm)
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1gA1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1u·A1gA2uA2gB1uB1gB2uB2gE2gE2uE1gE1u
A2g··A1gA1uB2gB2uB1gB1uE2uE2gE1uE1g
A2u···A1gB2uB2gB1uB1gE2gE2uE1gE1u
B1g····A1gA1uA2gA2uE1uE1gE2uE2g
B1u·····A1gA2uA2gE1gE1uE2gE2u
B2g······A1gA1uE1uE1gE2uE2g
B2u·······A1gE1gE1uE2gE2u
E2u········A1g+A2g+E2gA1u+A2u+E2uB1g+B2g+E1gB1u+B2u+E1u
E2g·········A1g+A2g+E2gB1u+B2u+E1uB1g+B2g+E1g
E1u··········A1g+A2g+E2gA1u+A2u+E2u
E1g···········A1g+A2g+E2g

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
[A1g x A1g]1···········
[A1u x A1u]1···········
[A2g x A2g]1···········
[A2u x A2u]1···········
[B1g x B1g]1···········
[B1u x B1u]1···········
[B2g x B2g]1···········
[B2u x B2u]1···········
[E2u x E2u]1········1··
[E2g x E2g]1········1··
[E1u x E1u]1········1··
[E1g x E1g]1········1··


Antisymmetrized Products of Irreps
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
{A1g x A1g}············
{A1u x A1u}············
{A2g x A2g}············
{A2u x A2u}············
{B1g x B1g}············
{B1u x B1u}············
{B2g x B2g}············
{B2u x B2u}············
{E2u x E2u}··1·········
{E2g x E2g}··1·········
{E1u x E1u}··1·········
{E1g x E1g}··1·········


Irreps Decompositions
D6h(6/mmm)A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
V···1······1·
[V2]2········1·1
[V3]···2·1·11·2·
[V4]3···1·1··3·2
A··1········1
[A2]2········1·1
[A3]··2·1·1··1·2
[A4]3···1·1··3·2
[V2]xV·1·3·1·12·4·
[[V2]2]5···1·1··4·3
{V2}··1········1
{A2}··1········1
{[V2]2}1·2·1·1··2·3

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRA1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1g···x······x·
A1u··x········x
A2g·x········x·
A2ux··········x
B1g·······xx···
B1u······x··x··
B2g·····x··x···
B2u····x····x··
E2u····x·x··x·x
E2g·····x·xx·x·
E1ux·x······x·x
E1g·x·x····x·x·

[ Note: x means allowed ]


Raman Selection Rules
RamanA1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
A1gx········x·x
A1u·x······x·x·
A2g··x······x·x
A2u···x····x·x·
B1g····x····x·x
B1u·····x··x·x·
B2g······x··x·x
B2u·······xx·x·
E2u·x·x·x·xx·x·
E2gx·x·x·x··x·x
E1u·x·x·x·xx·x·
E1gx·x·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 D6h(6/mmm)
L2L+1A1gA1uA2gA2uB1gB1uB2gB2uE2uE2gE1uE1g
011···········
13···1······1·
251········1·1
37···1·1·11·1·
491···1·1··2·1
511···1·1·12·2·
6132·1·1·1··2·2
715·1·2·1·12·3·
8172·1·1·1··3·3
919·1·2·2·23·3·
10212·1·2·2··4·3


* 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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