Point Group C29

Point Group C₂₉

ε=exp(2πi/29)
C₂₉	E	C₂₉	(C₂₉)²	(C₂₉)³	(C₂₉)⁴	(C₂₉)⁵	(C₂₉)⁶	(C₂₉)⁷	(C₂₉)⁸	(C₂₉)⁹	(C₂₉)¹⁰	(C₂₉)¹¹	(C₂₉)¹²	(C₂₉)¹³	(C₂₉)¹⁴	(C₂₉)¹⁵	(C₂₉)¹⁶	(C₂₉)¹⁷	(C₂₉)¹⁸	(C₂₉)¹⁹	(C₂₉)²⁰	(C₂₉)²¹	(C₂₉)²²	(C₂₉)²³	(C₂₉)²⁴	(C₂₉)²⁵	(C₂₉)²⁶	(C₂₉)²⁷	(C₂₉)²⁸
A	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
E₁*	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^6* ε^6*	ε^7* ε^7*	ε^8* ε^8*	ε^9* ε^9*	ε^10* ε^10*	ε^11* ε^11*	ε^12* ε^12*	ε^13* ε^13*	ε^14* ε^14*	ε^14* ε^14*	ε^13* ε^13*	ε^12* ε^12*	ε^11* ε^11*	ε^10* ε^10*	ε^9* ε^9*	ε^8* ε^8*	ε^7* ε^7*	ε^6* ε^6*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*
E₂*	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	ε^8* ε^8*	ε^10* ε^10*	ε^12* ε^12*	ε^14* ε^14*	ε^13* ε^13*	ε^11* ε^11*	ε^9* ε^9*	ε^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	ε^9* ε^9*	ε^11* ε^11*	ε^13* ε^13*	ε^14* ε^14*	ε^12* ε^12*	ε^10* ε^10*	ε^8* ε^8*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*
E₃*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^9* ε^9*	ε^12* ε^12*	ε^14* ε^14*	ε^11* ε^11*	ε^8* ε^8*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^7* ε^7*	ε^10* ε^10*	ε^13* ε^13*	ε^13* ε^13*	ε^10* ε^10*	ε^7* ε^7*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^8* ε^8*	ε^11* ε^11*	ε^14* ε^14*	ε^12* ε^12*	ε^9* ε^9*	ε^6* ε^6*	ε^3* ε^3*
E₄*	1 1	ε^4* ε^4*	ε^8* ε^8*	ε^12* ε^12*	ε^13* ε^13*	ε^9* ε^9*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^11* ε^11*	ε^14* ε^14*	ε^10* ε^10*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^10* ε^10*	ε^14* ε^14*	ε^11* ε^11*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^9* ε^9*	ε^13* ε^13*	ε^12* ε^12*	ε^8* ε^8*	ε^4* ε^4*
E₅*	1 1	ε^5* ε^5*	ε^10* ε^10*	ε^14* ε^14*	ε^9* ε^9*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	ε^11* ε^11*	ε^13* ε^13*	ε^8* ε^8*	ε^3* ε^3*	ε^2* ε^2*	ε^7* ε^7*	ε^12* ε^12*	ε^12* ε^12*	ε^7* ε^7*	ε^2* ε^2*	ε^3* ε^3*	ε^8* ε^8*	ε^13* ε^13*	ε^11* ε^11*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	ε^9* ε^9*	ε^14* ε^14*	ε^10* ε^10*	ε^5* ε^5*
E₆*	1 1	ε^6* ε^6*	ε^12* ε^12*	ε^11* ε^11*	ε^5* ε^5*	ε^* ε^*	ε^7* ε^7*	ε^13* ε^13*	ε^10* ε^10*	ε^4* ε^4*	ε^2* ε^2*	ε^8* ε^8*	ε^14* ε^14*	ε^9* ε^9*	ε^3* ε^3*	ε^3* ε^3*	ε^9* ε^9*	ε^14* ε^14*	ε^8* ε^8*	ε^2* ε^2*	ε^4* ε^4*	ε^10* ε^10*	ε^13* ε^13*	ε^7* ε^7*	ε^* ε^*	ε^5* ε^5*	ε^11* ε^11*	ε^12* ε^12*	ε^6* ε^6*
E₇*	1 1	ε^7* ε^7*	ε^14* ε^14*	ε^8* ε^8*	ε^* ε^*	ε^6* ε^6*	ε^13* ε^13*	ε^9* ε^9*	ε^2* ε^2*	ε^5* ε^5*	ε^12* ε^12*	ε^10* ε^10*	ε^3* ε^3*	ε^4* ε^4*	ε^11* ε^11*	ε^11* ε^11*	ε^4* ε^4*	ε^3* ε^3*	ε^10* ε^10*	ε^12* ε^12*	ε^5* ε^5*	ε^2* ε^2*	ε^9* ε^9*	ε^13* ε^13*	ε^6* ε^6*	ε^* ε^*	ε^8* ε^8*	ε^14* ε^14*	ε^7* ε^7*
E₈*	1 1	ε^8* ε^8*	ε^13* ε^13*	ε^5* ε^5*	ε^3* ε^3*	ε^11* ε^11*	ε^10* ε^10*	ε^2* ε^2*	ε^6* ε^6*	ε^14* ε^14*	ε^7* ε^7*	ε^* ε^*	ε^9* ε^9*	ε^12* ε^12*	ε^4* ε^4*	ε^4* ε^4*	ε^12* ε^12*	ε^9* ε^9*	ε^* ε^*	ε^7* ε^7*	ε^14* ε^14*	ε^6* ε^6*	ε^2* ε^2*	ε^10* ε^10*	ε^11* ε^11*	ε^3* ε^3*	ε^5* ε^5*	ε^13* ε^13*	ε^8* ε^8*
E₉*	1 1	ε^9* ε^9*	ε^11* ε^11*	ε^2* ε^2*	ε^7* ε^7*	ε^13* ε^13*	ε^4* ε^4*	ε^5* ε^5*	ε^14* ε^14*	ε^6* ε^6*	ε^3* ε^3*	ε^12* ε^12*	ε^8* ε^8*	ε^* ε^*	ε^10* ε^10*	ε^10* ε^10*	ε^* ε^*	ε^8* ε^8*	ε^12* ε^12*	ε^3* ε^3*	ε^6* ε^6*	ε^14* ε^14*	ε^5* ε^5*	ε^4* ε^4*	ε^13* ε^13*	ε^7* ε^7*	ε^2* ε^2*	ε^11* ε^11*	ε^9* ε^9*
E₁₀*	1 1	ε^10* ε^10*	ε^9* ε^9*	ε^* ε^*	ε^11* ε^11*	ε^8* ε^8*	ε^2* ε^2*	ε^12* ε^12*	ε^7* ε^7*	ε^3* ε^3*	ε^13* ε^13*	ε^6* ε^6*	ε^4* ε^4*	ε^14* ε^14*	ε^5* ε^5*	ε^5* ε^5*	ε^14* ε^14*	ε^4* ε^4*	ε^6* ε^6*	ε^13* ε^13*	ε^3* ε^3*	ε^7* ε^7*	ε^12* ε^12*	ε^2* ε^2*	ε^8* ε^8*	ε^11* ε^11*	ε^* ε^*	ε^9* ε^9*	ε^10* ε^10*
E₁₁*	1 1	ε^11* ε^11*	ε^7* ε^7*	ε^4* ε^4*	ε^14* ε^14*	ε^3* ε^3*	ε^8* ε^8*	ε^10* ε^10*	ε^* ε^*	ε^12* ε^12*	ε^6* ε^6*	ε^5* ε^5*	ε^13* ε^13*	ε^2* ε^2*	ε^9* ε^9*	ε^9* ε^9*	ε^2* ε^2*	ε^13* ε^13*	ε^5* ε^5*	ε^6* ε^6*	ε^12* ε^12*	ε^* ε^*	ε^10* ε^10*	ε^8* ε^8*	ε^3* ε^3*	ε^14* ε^14*	ε^4* ε^4*	ε^7* ε^7*	ε^11* ε^11*
E₁₂*	1 1	ε^12* ε^12*	ε^5* ε^5*	ε^7* ε^7*	ε^10* ε^10*	ε^2* ε^2*	ε^14* ε^14*	ε^3* ε^3*	ε^9* ε^9*	ε^8* ε^8*	ε^4* ε^4*	ε^13* ε^13*	ε^* ε^*	ε^11* ε^11*	ε^6* ε^6*	ε^6* ε^6*	ε^11* ε^11*	ε^* ε^*	ε^13* ε^13*	ε^4* ε^4*	ε^8* ε^8*	ε^9* ε^9*	ε^3* ε^3*	ε^14* ε^14*	ε^2* ε^2*	ε^10* ε^10*	ε^7* ε^7*	ε^5* ε^5*	ε^12* ε^12*
E₁₃*	1 1	ε^13* ε^13*	ε^3* ε^3*	ε^10* ε^10*	ε^6* ε^6*	ε^7* ε^7*	ε^9* ε^9*	ε^4* ε^4*	ε^12* ε^12*	ε^* ε^*	ε^14* ε^14*	ε^2* ε^2*	ε^11* ε^11*	ε^5* ε^5*	ε^8* ε^8*	ε^8* ε^8*	ε^5* ε^5*	ε^11* ε^11*	ε^2* ε^2*	ε^14* ε^14*	ε^* ε^*	ε^12* ε^12*	ε^4* ε^4*	ε^9* ε^9*	ε^7* ε^7*	ε^6* ε^6*	ε^10* ε^10*	ε^3* ε^3*	ε^13* ε^13*
E₁₄*	1 1	ε^14* ε^14*	ε^* ε^*	ε^13* ε^13*	ε^2* ε^2*	ε^12* ε^12*	ε^3* ε^3*	ε^11* ε^11*	ε^4* ε^4*	ε^10* ε^10*	ε^5* ε^5*	ε^9* ε^9*	ε^6* ε^6*	ε^8* ε^8*	ε^7* ε^7*	ε^7* ε^7*	ε^8* ε^8*	ε^6* ε^6*	ε^9* ε^9*	ε^5* ε^5*	ε^10* ε^10*	ε^4* ε^4*	ε^11* ε^11*	ε^3* ε^3*	ε^12* ε^12*	ε^2* ε^2*	ε^13* ε^13*	ε^* ε^*	ε^14* ε^14*

Additional information

Number of symmetry elements	h = 29
Number of classes, irreps	n = 29
Number of real-valued irreducible representations	n = 15
Abelian group	yes
Optical Isomerism (Chirality)	yes
Polar	yes
Parity	no

Reduce representation to irreducible representations

E	C₂₉	(C₂₉)²	(C₂₉)³	(C₂₉)⁴	(C₂₉)⁵	(C₂₉)⁶	(C₂₉)⁷	(C₂₉)⁸	(C₂₉)⁹	(C₂₉)¹⁰	(C₂₉)¹¹	(C₂₉)¹²	(C₂₉)¹³	(C₂₉)¹⁴	(C₂₉)¹⁵	(C₂₉)¹⁶	(C₂₉)¹⁷	(C₂₉)¹⁸	(C₂₉)¹⁹	(C₂₉)²⁰	(C₂₉)²¹	(C₂₉)²²	(C₂₉)²³	(C₂₉)²⁴	(C₂₉)²⁵	(C₂₉)²⁶	(C₂₉)²⁷	(C₂₉)²⁸

Genrate representation from irreducible representations

A	E₁*	E₂*	E₃*	E₄*	E₅*	E₆*	E₇*	E₈*	E₉*	E₁₀*	E₁₁*	E₁₂*	E₁₃*	E₁₄*

Direct products of irreducible representations

**Binary products**
⊙	A	E₁*	E₂*	E₃*	E₄*	E₅*	E₆*	E₇*	E₈*	E₉*	E₁₀*	E₁₁*	E₁₂*	E₁₃*	E₁₄*
A	A
E₁*	E₁	2A⊕E₂
E₂*	E₂	E₁⊕E₃	2A⊕E₄
E₃*	E₃	E₂⊕E₄	E₁⊕E₅	2A⊕E₆
E₄*	E₄	E₃⊕E₅	E₂⊕E₆	E₁⊕E₇	2A⊕E₈
E₅*	E₅	E₄⊕E₆	E₃⊕E₇	E₂⊕E₈	E₁⊕E₉	2A⊕E₁₀
E₆*	E₆	E₅⊕E₇	E₄⊕E₈	E₃⊕E₉	E₂⊕E₁₀	E₁⊕E₁₁	2A⊕E₁₂
E₇*	E₇	E₆⊕E₈	E₅⊕E₉	E₄⊕E₁₀	E₃⊕E₁₁	E₂⊕E₁₂	E₁⊕E₁₃	2A⊕E₁₄
E₈*	E₈	E₇⊕E₉	E₆⊕E₁₀	E₅⊕E₁₁	E₄⊕E₁₂	E₃⊕E₁₃	E₂⊕E₁₄	E₁⊕E₁₄	2A⊕E₁₃
E₉*	E₉	E₈⊕E₁₀	E₇⊕E₁₁	E₆⊕E₁₂	E₅⊕E₁₃	E₄⊕E₁₄	E₃⊕E₁₄	E₂⊕E₁₃	E₁⊕E₁₂	2A⊕E₁₁
E₁₀*	E₁₀	E₉⊕E₁₁	E₈⊕E₁₂	E₇⊕E₁₃	E₆⊕E₁₄	E₅⊕E₁₄	E₄⊕E₁₃	E₃⊕E₁₂	E₂⊕E₁₁	E₁⊕E₁₀	2A⊕E₉
E₁₁*	E₁₁	E₁₀⊕E₁₂	E₉⊕E₁₃	E₈⊕E₁₄	E₇⊕E₁₄	E₆⊕E₁₃	E₅⊕E₁₂	E₄⊕E₁₁	E₃⊕E₁₀	E₂⊕E₉	E₁⊕E₈	2A⊕E₇
E₁₂*	E₁₂	E₁₁⊕E₁₃	E₁₀⊕E₁₄	E₉⊕E₁₄	E₈⊕E₁₃	E₇⊕E₁₂	E₆⊕E₁₁	E₅⊕E₁₀	E₄⊕E₉	E₃⊕E₈	E₂⊕E₇	E₁⊕E₆	2A⊕E₅
E₁₃*	E₁₃	E₁₂⊕E₁₄	E₁₁⊕E₁₄	E₁₀⊕E₁₃	E₉⊕E₁₂	E₈⊕E₁₁	E₇⊕E₁₀	E₆⊕E₉	E₅⊕E₈	E₄⊕E₇	E₃⊕E₆	E₂⊕E₅	E₁⊕E₄	2A⊕E₃
E₁₄*	E₁₄	E₁₃⊕E₁₄	E₁₂⊕E₁₃	E₁₁⊕E₁₂	E₁₀⊕E₁₁	E₉⊕E₁₀	E₈⊕E₉	E₇⊕E₈	E₆⊕E₇	E₅⊕E₆	E₄⊕E₅	E₃⊕E₄	E₂⊕E₃	E₁⊕E₂	2A⊕E₁

Ternary Products
Quaternary Products

Symmetric powers [Γⁿ] of degenerate irreducible representations
Vibrational overtones

irrep	[Γ²]	[Γ³]	[Γ⁴]	[Γ⁵]	[Γ⁶]
E₁*	A⊕E₂	E₁⊕E₃	A⊕E₂⊕E₄	E₁⊕E₃⊕E₅	A⊕E₂⊕E₄⊕E₆	More
E₂*	A⊕E₄	E₂⊕E₆	A⊕E₄⊕E₈	E₂⊕E₆⊕E₁₀	A⊕E₄⊕E₈⊕E₁₂	More
E₃*	A⊕E₆	E₃⊕E₉	A⊕E₆⊕E₁₂	E₃⊕E₉⊕E₁₄	A⊕E₆⊕E₁₁⊕E₁₂	More
E₄*	A⊕E₈	E₄⊕E₁₂	A⊕E₈⊕E₁₃	E₄⊕E₉⊕E₁₂	A⊕E₅⊕E₈⊕E₁₃	More
E₅*	A⊕E₁₀	E₅⊕E₁₄	A⊕E₉⊕E₁₀	E₄⊕E₅⊕E₁₄	A⊕E₁⊕E₉⊕E₁₀	More
E₆*	A⊕E₁₂	E₆⊕E₁₁	A⊕E₅⊕E₁₂	E₁⊕E₆⊕E₁₁	A⊕E₅⊕E₇⊕E₁₂	More
E₇*	A⊕E₁₄	E₇⊕E₈	A⊕E₁⊕E₁₄	E₆⊕E₇⊕E₈	A⊕E₁⊕E₁₃⊕E₁₄	More
E₈*	A⊕E₁₃	E₅⊕E₈	A⊕E₃⊕E₁₃	E₅⊕E₈⊕E₁₁	A⊕E₃⊕E₁₀⊕E₁₃	More
E₉*	A⊕E₁₁	E₂⊕E₉	A⊕E₇⊕E₁₁	E₂⊕E₉⊕E₁₃	A⊕E₄⊕E₇⊕E₁₁	More
E₁₀*	A⊕E₉	E₁⊕E₁₀	A⊕E₉⊕E₁₁	E₁⊕E₈⊕E₁₀	A⊕E₂⊕E₉⊕E₁₁	More
E₁₁*	A⊕E₇	E₄⊕E₁₁	A⊕E₇⊕E₁₄	E₃⊕E₄⊕E₁₁	A⊕E₇⊕E₈⊕E₁₄	More
E₁₂*	A⊕E₅	E₇⊕E₁₂	A⊕E₅⊕E₁₀	E₂⊕E₇⊕E₁₂	A⊕E₅⊕E₁₀⊕E₁₄	More
E₁₃*	A⊕E₃	E₁₀⊕E₁₃	A⊕E₃⊕E₆	E₇⊕E₁₀⊕E₁₃	A⊕E₃⊕E₆⊕E₉	More
E₁₄*	A⊕E₁	E₁₃⊕E₁₄	A⊕E₁⊕E₂	E₁₂⊕E₁₃⊕E₁₄	A⊕E₁⊕E₂⊕E₃	More

Spherical harmonics and Multipoles
Symmetric Powers of Γ_xyz

Spherical Harmonics Y^l / Multipole				Symmetric Power [Γ^l(xyz)]
l	2l+1	Multipole	Symmetry	Rank	[Γ^l(xyz)]
s (l=0)	1	Monopole	A	1	A
p (l=1)	3	Dipole	A⊕E₁	3	A⊕E₁
d (l=2)	5	Quadrupole	A⊕E₁⊕E₂	6	2A⊕E₁⊕E₂
f (l=3)	7	Octupole	A⊕E₁⊕E₂⊕E₃	10	2A⊕2E₁⊕E₂⊕E₃
g (l=4)	9	Hexadecapole	A⊕E₁⊕E₂⊕E₃⊕E₄	15	3A⊕2E₁⊕2E₂⊕E₃⊕E₄
h (l=5)	11	Dotricontapole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅	21	3A⊕3E₁⊕2E₂⊕2E₃⊕E₄⊕E₅
i (l=6)	13	Tetrahexacontapole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆	28	4A⊕3E₁⊕3E₂⊕2E₃⊕2E₄⊕E₅⊕E₆
j (l=7)	15	Octacosahectapole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇	36	4A⊕4E₁⊕3E₂⊕3E₃⊕2E₄⊕2E₅⊕E₆⊕E₇
k (l=8)	17	256-pole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈	45	5A⊕4E₁⊕4E₂⊕3E₃⊕3E₄⊕2E₅⊕2E₆⊕E₇⊕E₈
l (l=9)	19	512-pole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉	55	5A⊕5E₁⊕4E₂⊕4E₃⊕3E₄⊕3E₅⊕2E₆⊕2E₇⊕E₈⊕E₉
m (l=10)	21	1024-pole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀	66	6A⊕5E₁⊕5E₂⊕4E₃⊕4E₄⊕3E₅⊕3E₆⊕2E₇⊕2E₈⊕E₉⊕E₁₀
n (l=11)	23	2048-pole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀⊕E₁₁	78	6A⊕6E₁⊕5E₂⊕5E₃⊕4E₄⊕4E₅⊕3E₆⊕3E₇⊕2E₈⊕2E₉⊕E₁₀⊕E₁₁
o (l=12)	25	4096-pole	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀⊕E₁₁⊕E₁₂	91	7A⊕6E₁⊕6E₂⊕5E₃⊕5E₄⊕4E₅⊕4E₆⊕3E₇⊕3E₈⊕2E₉⊕2E₁₀⊕E₁₁⊕E₁₂
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First nonvanshing multipole: Dipole

Further Reading

A. Gelessus, W. Thiel, W. Weber. J. Chem. Educ. 72 505 (1995)
Multipoles and symmetry

Ligand Field, dⁿ term splitting

**Term symbols for electronic configurations dⁿ**
dⁿ	Term Symbols
d¹ = d⁹	²D
d² = d⁸	¹S, ¹D, ¹G, ³P, ³F
d³ = d⁷	²P, ²D (2), ²F, ²G, ²H, ⁴P, ⁴F
d⁴ = d⁶	¹S (2), ¹D (2), ¹F, ¹G (2), ¹I, ³P (2), ³D, ³F (2), ³G, ³H, ⁵D
d⁵	²S, ²P, ²D (3), ²F (2), ²G (2), ²H, ²I, ⁴P, ⁴D, ⁴F, ⁴G, ⁶S

**Term splitting in point group C₂₉**
L	2L+1	Term Splitting
S (L=0)	1	A
P (L=1)	3	A⊕E₁
D (L=2)	5	A⊕E₁⊕E₂
F (L=3)	7	A⊕E₁⊕E₂⊕E₃
G (L=4)	9	A⊕E₁⊕E₂⊕E₃⊕E₄
H (L=5)	11	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅
I (L=6)	13	A⊕E₁⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆

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