Point Group S29

Point Group S₂₉

ε=exp(2πi/29)
S₂₉	E	S₂₉	(C₂₉)²	(S₂₉)³	(C₂₉)⁴	(S₂₉)⁵	(C₂₉)⁶	(S₂₉)⁷	(C₂₉)⁸	(S₂₉)⁹	(C₂₉)¹⁰	(S₂₉)¹¹	(C₂₉)¹²	(S₂₉)¹³	(C₂₉)¹⁴	(S₂₉)¹⁵	(C₂₉)¹⁶	(S₂₉)¹⁷	(C₂₉)¹⁸	(S₂₉)¹⁹	(C₂₉)²⁰	(S₂₉)²¹	(C₂₉)²²	(S₂₉)²³	(C₂₉)²⁴	(S₂₉)²⁵	(C₂₉)²⁶	(S₂₉)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
B	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^6* -ε^6*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^* -ε^*		-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^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^6* -ε^6*	-ε^4* -ε^4*	-ε^2* -ε^2*		-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^6* -ε^6*	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^5* -ε^5*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^5* -ε^5*		ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*
E₃*	1 1	ε^3* ε^3*	ε^6* ε^6*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*		ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*		-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-ε^5* -ε^5*	ε^7* ε^7*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	-ε^6* -ε^6*	-ε^3* -ε^3*		-ε^3* -ε^3*	-ε^6* -ε^6*	ε^6* ε^6*	ε^3* ε^3*
E₄*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*		-ε^3* -ε^3*	-ε^* -ε^*	-ε^5* -ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	-ε^4* -ε^4*		-ε^4* -ε^4*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	-ε^5* -ε^5*	-ε^* -ε^*	-ε^3* -ε^3*		ε^4* ε^4*
E₅*	1 1	ε^5* ε^5*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^6* -ε^6*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	-ε^3* -ε^3*	-ε^2* -ε^2*		ε^3* ε^3*	ε^2* ε^2*		-ε^2* -ε^2*	-ε^3* -ε^3*	ε^7* ε^7*	ε^2* ε^2*	ε^3* ε^3*	-ε^6* -ε^6*	-ε^* -ε^*	-ε^4* -ε^4*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	-ε^5* -ε^5*		-ε^5* -ε^5*	ε^5* ε^5*
E₆*	1 1	ε^6* ε^6*	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^5* ε^5*	ε^* ε^*		-ε^* -ε^*	-ε^5* -ε^5*	ε^4* ε^4*	ε^2* ε^2*	-ε^6* -ε^6*		-ε^6* -ε^6*	ε^3* ε^3*	ε^3* ε^3*	-ε^5* -ε^5*	-ε^* -ε^*		ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^7* ε^7*	ε^* ε^*	ε^5* ε^5*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*
E₇*	1 1				ε^* ε^*	ε^6* ε^6*	-ε^* -ε^*	-ε^6* -ε^6*	ε^2* ε^2*	ε^5* ε^5*	-ε^2* -ε^2*	-ε^5* -ε^5*	ε^3* ε^3*	ε^4* ε^4*	-ε^3* -ε^3*	-ε^4* -ε^4*	ε^4* ε^4*	ε^3* ε^3*	-ε^4* -ε^4*	-ε^3* -ε^3*	ε^5* ε^5*	ε^2* ε^2*	-ε^5* -ε^5*	-ε^2* -ε^2*	ε^6* ε^6*	ε^* ε^*	-ε^6* -ε^6*	-ε^* -ε^*	ε^7* ε^7*
E₈*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^5* ε^5*	ε^3* ε^3*	-ε^3* -ε^3*	-ε^5* -ε^5*	ε^2* ε^2*	ε^6* ε^6*		ε^7* ε^7*	ε^* ε^*	-ε^5* -ε^5*	-ε^3* -ε^3*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^* ε^*		-ε^* -ε^*	ε^6* ε^6*	ε^2* ε^2*	-ε^4* -ε^4*	-ε^4* -ε^4*	ε^3* ε^3*	ε^5* ε^5*	-ε^* -ε^*
E₉*	1 1	-ε^5* -ε^5*	-ε^4* -ε^4*	ε^2* ε^2*		-ε^2* -ε^2*	ε^4* ε^4*	ε^5* ε^5*		ε^6* ε^6*	ε^3* ε^3*	-ε^2* -ε^2*		ε^* ε^*	-ε^4* -ε^4*	-ε^5* -ε^5*	ε^* ε^*	-ε^6* -ε^6*	-ε^3* -ε^3*	ε^3* ε^3*	ε^6* ε^6*	-ε^* -ε^*	ε^5* ε^5*	ε^4* ε^4*	-ε^* -ε^*	ε^7* ε^7*	ε^2* ε^2*	-ε^3* -ε^3*	-ε^6* -ε^6*
E₁₀*	1 1	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^* ε^*	-ε^3* -ε^3*		ε^2* ε^2*	-ε^2* -ε^2*	ε^7* ε^7*	ε^3* ε^3*	-ε^* -ε^*	ε^6* ε^6*	ε^4* ε^4*		ε^5* ε^5*	ε^5* ε^5*	-ε^* -ε^*	ε^4* ε^4*	ε^6* ε^6*	-ε^2* -ε^2*	ε^3* ε^3*		-ε^3* -ε^3*	ε^2* ε^2*	-ε^6* -ε^6*	-ε^4* -ε^4*	ε^* ε^*	-ε^5* -ε^5*	-ε^5* -ε^5*
E₁₁*	1 1	-ε^3* -ε^3*	ε^7* ε^7*	ε^4* ε^4*	-ε^* -ε^*	ε^3* ε^3*	-ε^6* -ε^6*	-ε^5* -ε^5*	ε^* ε^*	-ε^2* -ε^2*	ε^6* ε^6*	ε^5* ε^5*	-ε^2* -ε^2*	ε^2* ε^2*	-ε^5* -ε^5*	-ε^6* -ε^6*	ε^2* ε^2*	-ε^* -ε^*	ε^5* ε^5*	ε^6* ε^6*	-ε^3* -ε^3*	ε^* ε^*	-ε^4* -ε^4*		ε^3* ε^3*		ε^4* ε^4*		-ε^4* -ε^4*
E₁₂*	1 1	-ε^2* -ε^2*	ε^5* ε^5*		-ε^5* -ε^5*	ε^2* ε^2*		ε^3* ε^3*	-ε^5* -ε^5*		ε^4* ε^4*	-ε^2* -ε^2*	ε^* ε^*	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^4* -ε^4*	ε^* ε^*	-ε^* -ε^*	ε^4* ε^4*	-ε^6* -ε^6*	-ε^6* -ε^6*	ε^3* ε^3*	-ε^* -ε^*	ε^2* ε^2*	-ε^4* -ε^4*	ε^7* ε^7*	ε^5* ε^5*	-ε^3* -ε^3*
E₁₃*	1 1	-ε^* -ε^*	ε^3* ε^3*	-ε^4* -ε^4*	ε^6* ε^6*		-ε^6* -ε^6*	ε^4* ε^4*	-ε^3* -ε^3*	ε^* ε^*		ε^2* ε^2*	-ε^3* -ε^3*	ε^5* ε^5*	-ε^6* -ε^6*		ε^5* ε^5*	-ε^4* -ε^4*	ε^2* ε^2*	-ε^* -ε^*	ε^* ε^*	-ε^2* -ε^2*	ε^4* ε^4*	-ε^5* -ε^5*	ε^7* ε^7*	ε^6* ε^6*	-ε^5* -ε^5*	ε^3* ε^3*	-ε^2* -ε^2*

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)	no
Polar	no
Parity	no

Reduce representation to irreducible representations

E	S₂₉	(C₂₉)²	(S₂₉)³	(C₂₉)⁴	(S₂₉)⁵	(C₂₉)⁶	(S₂₉)⁷	(C₂₉)⁸	(S₂₉)⁹	(C₂₉)¹⁰	(S₂₉)¹¹	(C₂₉)¹²	(S₂₉)¹³	(C₂₉)¹⁴	(S₂₉)¹⁵	(C₂₉)¹⁶	(S₂₉)¹⁷	(C₂₉)¹⁸	(S₂₉)¹⁹	(C₂₉)²⁰	(S₂₉)²¹	(C₂₉)²²	(S₂₉)²³	(C₂₉)²⁴	(S₂₉)²⁵	(C₂₉)²⁶	(S₂₉)²⁷	(C₂₉)²⁸

Genrate representation from irreducible representations

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

Direct products of irreducible representations

Error: Reduction failed

Reduction to integer numbers for irreducible representations failed.
Error Code: 1

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

**Characters of reducible representation**
E	S₂₉	(C₂₉)²	(S₂₉)³	(C₂₉)⁴	(S₂₉)⁵	(C₂₉)⁶	(S₂₉)⁷	(C₂₉)⁸	(S₂₉)⁹	(C₂₉)¹⁰	(S₂₉)¹¹	(C₂₉)¹²	(S₂₉)¹³	(C₂₉)¹⁴	(S₂₉)¹⁵	(C₂₉)¹⁶	(S₂₉)¹⁷	(C₂₉)¹⁸	(S₂₉)¹⁹	(C₂₉)²⁰	(S₂₉)²¹	(C₂₉)²²	(S₂₉)²³	(C₂₉)²⁴	(S₂₉)²⁵	(C₂₉)²⁶	(S₂₉)²⁷	(C₂₉)²⁸
1	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000	-1.000	1.000