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Point Group Tables of C6(6)

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Character Table of the group C6(6)*
C6(6)#16+3+23-6-functions
AΓ1111111z,x2+y2,z2,Jz
BΓ41-11-11-1·
1E2
2E2		Γ3
Γ2		1
1		w
w2		w2
w		1
1		w
w2		w2
w		(x2-y2,xy)
2E1
1E1		Γ5
Γ6		1
1		-w2
-w		w
w2		-1
-1		w2
w		-w
-w2		(x,y),(xz,yz),(Jx,Jy)
[Note: there are differences between Bradley & Cracknell (1972) and Altmann & Herzig (1994)]

w = exp(2iπ/3)



Subgroups of the group C6(6)
SubgroupOrderIndex
C6(6)61
C3(3)32
C2(2)23
C1(1)16

[ Subduction tables ]

Multiplication Table of irreducible representations of the group C6(6)
C6(6)AB2E21E11E22E1
AAB2E21E11E22E1
B·A1E12E22E11E2
2E2··1E22E1AB
1E1···1E2BA
1E2····2E21E1
2E1·····2E2

[ Note: the table is symmetric ]


Symmetrized Products of Irreps
C6(6)AB2E21E11E22E1
[A x A]1·····
[B x B]1·····
[2E2 x 2E2]····1·
[1E1 x 1E1]····1·
[1E2 x 1E2]··1···
[2E1 x 2E1]··1···


Antisymmetrized Products of Irreps
C6(6)AB2E21E11E22E1
{A x A}······
{B x B}······
{2E2 x 2E2}······
{1E1 x 1E1}······
{1E2 x 1E2}······
{2E1 x 2E1}······


Irreps Decompositions
C6(6)AB2E21E11E22E1
V1··1·1
[V2]2·1111
[V3]221212
[V4]323232
A1··1·1
[A2]2·1111
[A3]221212
[A4]323232
[V2]xV422424
[[V2]2]524343
{V2}1··1·1
{A2}1··1·1
{[V2]2}322323

V ≡ the vector representation
A ≡ the axial representation


IR Selection Rules
IRAB2E21E11E22E1
Ax··x·x
B·xx·x·
2E2·xx··x
1E1x··xx·
1E2·x·xx·
2E1x·x··x

[ Note: x means allowed ]


Raman Selection Rules
RamanAB2E21E11E22E1
Ax·xxxx
B·xxxxx
2E2xxx·xx
1E1xx·xxx
1E2xxxxx·
2E1xxxx·x

[ Note: x means allowed ]


Irreps Dimensions Irreps of the point group
Subduction of the rotation group D(L) to irreps of the group C6(6)
L2L+1AB2E21E11E22E1
011·····
131··1·1
251·1111
37121111
49122121
511122222
613322222
715322323
817323333
919343333
1021344343


* 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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