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Point Group Tables of C6h(6/m)

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Character Table of the group C6h(6/m)*
C6h(6/m)#16+3+23-6--1-6+-3+m-3--6-functions
AgΓ1+111111111111x2+y2,z2,Jz
AuΓ1-111111-1-1-1-1-1-1z
BgΓ4+1-11-11-11-11-11-1·
BuΓ4-1-11-11-1-11-11-11·
1E2g
2E2g		Γ3+
Γ2+		1
1		w
w2		w2
w		1
1		w
w2		w2
w		1
1		w
w2		w2
w		1
1		w
w2		w2
w		(x2-y2,xy)
1E2u
2E2u		Γ3-
Γ2-		1
1		w
w2		w2
w		1
1		w
w2		w2
w		-1
-1		-w
-w2		-w2
-w		-1
-1		-w
-w2		-w2
-w		·
2E1g
1E1g		Γ5+
Γ6+		1
1		-w2
-w		w
w2		-1
-1		w2
w		-w
-w2		1
1		-w2
-w		w
w2		-1
-1		w2
w		-w
-w2		(xz,yz),(Jx,Jy)
2E1u
1E1u		Γ5-
Γ6-		1
1		-w2
-w		w
w2		-1
-1		w2
w		-w
-w2		-1
-1		w2
w		-w
-w2		1
1		-w2
-w		w
w2		(x,y)

w = exp(2iπ/3)



Subgroups of the group C6h(6/m)
SubgroupOrderIndex
C6h(6/m)121
C3h(-6)62
C6(6)62
C3i(-3)62
C3(3)34
C2h(2/m)43
Cs(m)26
C2(2)26
Ci(-1)26
C1(1)112

[ Subduction tables ]

Multiplication Table of irreducible representations of the group C6h(6/m)
C6h(6/m)AgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
AgAgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
Au·AgBuBg2E2u2E2g1E1u1E1g1E2u1E2g2E1u2E1g
Bg··AgAu1E1g1E1u2E2g2E2u2E1g2E1u1E2g1E2u
Bu···Ag1E1u1E1g2E2u2E2g2E1u2E1g1E2u1E2g
2E2g····1E2g1E2u2E1g2E1uAgAuBgBu
2E2u·····1E2g2E1u2E1gAuAgBuBg
1E1g······1E2g1E2uBgBuAgAu
1E1u·······1E2gBuBgAuAg
1E2g········2E2g2E2u1E1g1E1u
1E2u·········2E2g1E1u1E1g
2E1g··········2E2g2E2u
2E1u···········2E2g

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
C6h(6/m)AgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
[Ag x Ag]1···········
[Au x Au]1···········
[Bg x Bg]1···········
[Bu x Bu]1···········
[2E2g x 2E2g]········1···
[2E2u x 2E2u]········1···
[1E1g x 1E1g]········1···
[1E1u x 1E1u]········1···
[1E2g x 1E2g]····1·······
[1E2u x 1E2u]····1·······
[2E1g x 2E1g]····1·······
[2E1u x 2E1u]····1·······


Antisymmetrized Products of Irreps
C6h(6/m)AgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
{Ag x Ag}············
{Au x Au}············
{Bg x Bg}············
{Bu x Bu}············
{2E2g x 2E2g}············
{2E2u x 2E2u}············
{1E1g x 1E1g}············
{1E1u x 1E1u}············
{1E2g x 1E2g}············
{1E2u x 1E2u}············
{2E1g x 2E1g}············
{2E1u x 2E1u}············


Irreps Decompositions
C6h(6/m)AgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
V·1·····1···1
[V2]2···1·1·1·1·
[V3]·2·2·1·2·1·2
[V4]3·2·3·2·3·2·
A1·····1···1·
[A2]2···1·1·1·1·
[A3]2·2·1·2·1·2·
[A4]3·2·3·2·3·2·
[V2]xV·4·2·2·4·2·4
[[V2]2]5·2·4·3·4·3·
{V2}1·····1···1·
{A2}1·····1···1·
{[V2]2}3·2·2·3·2·3·

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRAgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
Ag·x·····x···x
Aux·····x···x·
Bg···x·x···x··
Bu··x·x···x···
2E2g···x·x·····x
2E2u··x·x·····x·
1E1g·x·····x·x··
1E1ux·····x·x···
1E2g···x···x·x··
1E2u··x···x·x···
2E1g·x···x·····x
2E1ux···x·····x·

[ Note: x means allowed ]


Raman Selection Rules
RamanAgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
Agx···x·x·x·x·
Au·x···x·x·x·x
Bg··x·x·x·x·x·
Bu···x·x·x·x·x
2E2gx·x·x···x·x·
2E2u·x·x·x···x·x
1E1gx·x···x·x·x·
1E1u·x·x···x·x·x
1E2gx·x·x·x·x···
1E2u·x·x·x·x·x··
2E1gx·x·x·x···x·
2E1u·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 C6h(6/m)
L2L+1AgAuBgBu2E2g2E2u1E1g1E1u1E2g1E2u2E1g2E1u
011···········
13·1·····1···1
251···1·1·1·1·
37·1·2·1·1·1·1
491·2·2·1·2·1·
511·1·2·2·2·2·2
6133·2·2·2·2·2·
715·3·2·2·3·2·3
8173·2·3·3·3·3·
919·3·4·3·3·3·3
10213·4·4·3·4·3·


* 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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