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Point Group Tables of C_3h(-6)

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Character Table of the group C_3h(-6)*
C_3h(-6)	#	1	3⁺	3^-	m	-6^-	-6⁺	functions
A^'	Γ₁	1	1	1	1	1	1	x²+y²,z²,J_z
A^''	Γ₄	1	1	1	-1	-1	-1	z
²E^' ¹E^'	Γ₂ Γ₃	1 1	w w²	w² w	1 1	w w²	w² w	(x,y),(x²-y²,xy)
²E^'' ¹E^''	Γ₅ Γ₆	1 1	w w²	w² w	-1 -1	-w -w²	-w² -w	(xz,yz),(J_x,J_y)

w = exp(2iπ/3)

**Subgroups of the group C_3h(-6)**
Subgroup	Order	Index
C_3h(-6)	6	1
C₃(3)	3	2
C_s(m)	2	3
C₁(1)	1	6

[ Subduction tables ]

**Multiplication Table of irreducible representations of the group C_3h(-6)**
C_3h(-6)	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
A^'	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
A^''	·	A^'	¹E^''	¹E^'	²E^''	²E^'
¹E^'	·	·	²E^'	²E^''	A^'	A^''
¹E^''	·	·	·	²E^'	A^''	A^'
²E^'	·	·	·	·	¹E^'	¹E^''
²E^''	·	·	·	·	·	¹E^'

[ Note: the table is symmetric ]

**Symmetrized Products of Irreps**
C_3h(-6)	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
[A^' x A^']	1	·	·	·	·	·
[A^'' x A^'']	1	·	·	·	·	·
[¹E^' x ¹E^']	·	·	·	·	1	·
[¹E^'' x ¹E^'']	·	·	·	·	1	·
[²E^' x ²E^']	·	·	1	·	·	·
[²E^'' x ²E^'']	·	·	1	·	·	·

**Antisymmetrized Products of Irreps**
C_3h(-6)	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
{A^' x A^'}	·	·	·	·	·	·
{A^'' x A^''}	·	·	·	·	·	·
{¹E^' x ¹E^'}	·	·	·	·	·	·
{¹E^'' x ¹E^''}	·	·	·	·	·	·
{²E^' x ²E^'}	·	·	·	·	·	·
{²E^'' x ²E^''}	·	·	·	·	·	·

**Irreps Decompositions**
C_3h(-6)	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
V	·	1	1	·	1	·
[V²]	2	·	1	1	1	1
[V³]	2	2	2	1	2	1
[V⁴]	3	2	3	2	3	2
A	1	·	·	1	·	1
[A²]	2	·	1	1	1	1
[A³]	2	2	1	2	1	2
[A⁴]	3	2	3	2	3	2
[V²]xV	2	4	4	2	4	2
[[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**
IR	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
A^'	·	x	x	·	x	·
A^''	x	·	·	x	·	x
¹E^'	x	·	·	x	x	·
¹E^''	·	x	x	·	·	x
²E^'	x	·	x	·	·	x
²E^''	·	x	·	x	x	·

[ Note: x means allowed ]

**Raman Selection Rules**
Raman	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
A^'	x	·	x	x	x	x
A^''	·	x	x	x	x	x
¹E^'	x	x	x	·	x	x
¹E^''	x	x	·	x	x	x
²E^'	x	x	x	x	x	·
²E^''	x	x	x	x	·	x

[ Note: x means allowed ]

**Subduction of the rotation group D(L) to irreps of the group C_3h(-6)**
Irreps	Dimensions	Irreps of the point group
L	2L+1	A^'	A^''	¹E^'	¹E^''	²E^'	²E^''
0	1	1	·	·	·	·	·
1	3	·	1	1	·	1	·
2	5	1	·	1	1	1	1
3	7	2	1	1	1	1	1
4	9	1	2	2	1	2	1
5	11	2	1	2	2	2	2
6	13	3	2	2	2	2	2
7	15	2	3	3	2	3	2
8	17	3	2	3	3	3	3
9	19	4	3	3	3	3	3
10	21	3	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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Point Group Tables of C3h(-6)

Point Group Tables of C_3h(-6)