Point Group C30

Point Group C₃₀

ε=exp(2πi/30)
C₃₀	E	C₃₀	C₁₅	C₁₀	(C₁₅)²	C₆	C₅	(C₃₀)⁷	(C₁₅)⁴	(C₁₀)³	C₃	(C₃₀)¹¹	(C₅)²	(C₃₀)¹³	(C₁₅)⁷	C₂	(C₁₅)⁸	(C₃₀)¹⁷	(C₅)³	(C₃₀)¹⁹	(C₃)²	(C₁₀)⁷	(C₁₅)¹¹	(C₃₀)²³	(C₅)⁴	(C₆)⁵	(C₁₅)¹³	(C₁₀)⁹	(C₁₅)¹⁴	(C₃₀)²⁹
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	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	-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^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^*	-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^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^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^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*	1 1	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^7* -ε^7*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*
E₃*	1 1	ε^3* ε^3*	ε^6* ε^6*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^6* ε^6*	ε^3* ε^3*
E₄*	1 1	ε^4* ε^4*	-ε^7* -ε^7*	-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^5* -ε^5*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^7* -ε^7*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^7* -ε^7*	-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^5* -ε^5*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^7* -ε^7*	ε^4* ε^4*
E₅*	1 1	ε^5* ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^5* ε^5*	1 1	ε^5* ε^5*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^5* ε^5*
E₆*	1 1	ε^6* ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^3* -ε^3*	-ε^3* -ε^3*	ε^6* ε^6*
E₇*	1 1	ε^7* ε^7*	-ε^* -ε^*	-ε^6* -ε^6*	ε^2* ε^2*	ε^5* ε^5*	-ε^3* -ε^3*	-ε^4* -ε^4*	ε^4* ε^4*	ε^3* ε^3*	-ε^5* -ε^5*	-ε^2* -ε^2*	ε^6* ε^6*	ε^* ε^*	-ε^7* -ε^7*	-1 -1	-ε^7* -ε^7*	ε^* ε^*	ε^6* ε^6*	-ε^2* -ε^2*	-ε^5* -ε^5*	ε^3* ε^3*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^3* -ε^3*	ε^5* ε^5*	ε^2* ε^2*	-ε^6* -ε^6*	-ε^* -ε^*	ε^7* ε^7*
E₈*	1 1	-ε^7* -ε^7*	-ε^* -ε^*	ε^6* ε^6*	ε^2* ε^2*	-ε^5* -ε^5*	-ε^3* -ε^3*	ε^4* ε^4*	ε^4* ε^4*	-ε^3* -ε^3*	-ε^5* -ε^5*	ε^2* ε^2*	ε^6* ε^6*	-ε^* -ε^*	-ε^7* -ε^7*	1 1	-ε^7* -ε^7*	-ε^* -ε^*	ε^6* ε^6*	ε^2* ε^2*	-ε^5* -ε^5*	-ε^3* -ε^3*	ε^4* ε^4*	ε^4* ε^4*	-ε^3* -ε^3*	-ε^5* -ε^5*	ε^2* ε^2*	ε^6* ε^6*	-ε^* -ε^*	-ε^7* -ε^7*
E₉*	1 1	-ε^6* -ε^6*	-ε^3* -ε^3*	ε^3* ε^3*	ε^6* ε^6*	-1 -1	ε^6* ε^6*	ε^3* ε^3*	-ε^3* -ε^3*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^3* -ε^3*	ε^3* ε^3*	ε^6* ε^6*	-1 -1	ε^6* ε^6*	ε^3* ε^3*	-ε^3* -ε^3*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^3* -ε^3*	ε^3* ε^3*	ε^6* ε^6*	-1 -1	ε^6* ε^6*	ε^3* ε^3*	-ε^3* -ε^3*	-ε^6* -ε^6*
E₁₀*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*	1 1	-ε^5* -ε^5*	-ε^5* -ε^5*
E₁₁*	1 1	-ε^4* -ε^4*	-ε^7* -ε^7*	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^5* -ε^5*	ε^* ε^*	-ε^3* -ε^3*	ε^7* ε^7*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^7* ε^7*	-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^5* ε^5*	-ε^* -ε^*	ε^3* ε^3*	-ε^7* -ε^7*	-ε^4* -ε^4*
E₁₂*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^3* -ε^3*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^6* ε^6*	-ε^3* -ε^3*
E₁₃*	1 1	-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^7* ε^7*	ε^6* ε^6*	-ε^4* -ε^4*	ε^2* ε^2*	-1 -1	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^7* -ε^7*	-ε^6* -ε^6*	ε^4* ε^4*	-ε^2* -ε^2*
E₁₄*	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^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^*	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^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^*

Additional information

Number of symmetry elements	h = 30
Number of classes, irreps	n = 30
Number of real-valued irreducible representations	n = 16
Abelian group	yes
Optical Isomerism (Chirality)	yes
Polar	yes
Parity	no

Reduce representation to irreducible representations

E	C₃₀	C₁₅	C₁₀	(C₁₅)²	C₆	C₅	(C₃₀)⁷	(C₁₅)⁴	(C₁₀)³	C₃	(C₃₀)¹¹	(C₅)²	(C₃₀)¹³	(C₁₅)⁷	C₂	(C₁₅)⁸	(C₃₀)¹⁷	(C₅)³	(C₃₀)¹⁹	(C₃)²	(C₁₀)⁷	(C₁₅)¹¹	(C₃₀)²³	(C₅)⁴	(C₆)⁵	(C₁₅)¹³	(C₁₀)⁹	(C₁₅)¹⁴	(C₃₀)²⁹

Genrate representation from irreducible representations

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

Direct products of irreducible representations

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

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	2B⊕E₃⊕E₉	A⊕E₆⊕2E₁₂	More
E₄*	A⊕E₈	E₄⊕E₁₂	A⊕E₈⊕E₁₄	E₄⊕E₁₀⊕E₁₂	A⊕E₆⊕E₈⊕E₁₄	More
E₅*	A⊕E₁₀	2B⊕E₅	A⊕2E₁₀	2B⊕2E₅	3A⊕2E₁₀	More
E₆*	A⊕E₁₂	E₆⊕E₁₂	A⊕E₆⊕E₁₂	2A⊕E₆⊕E₁₂	A⊕2E₆⊕E₁₂	More
E₇*	A⊕E₁₄	E₇⊕E₉	A⊕E₂⊕E₁₄	E₅⊕E₇⊕E₉	A⊕E₂⊕E₁₂⊕E₁₄	More
E₈*	A⊕E₁₄	E₆⊕E₈	A⊕E₂⊕E₁₄	E₆⊕E₈⊕E₁₀	A⊕E₂⊕E₁₂⊕E₁₄	More
E₉*	A⊕E₁₂	E₃⊕E₉	A⊕E₆⊕E₁₂	2B⊕E₃⊕E₉	A⊕2E₆⊕E₁₂	More
E₁₀*	A⊕E₁₀	2A⊕E₁₀	A⊕2E₁₀	2A⊕2E₁₀	3A⊕2E₁₀	More
E₁₁*	A⊕E₈	E₃⊕E₁₁	A⊕E₈⊕E₁₄	E₃⊕E₅⊕E₁₁	A⊕E₆⊕E₈⊕E₁₄	More
E₁₂*	A⊕E₆	E₆⊕E₁₂	A⊕E₆⊕E₁₂	2A⊕E₆⊕E₁₂	A⊕E₆⊕2E₁₂	More
E₁₃*	A⊕E₄	E₉⊕E₁₃	A⊕E₄⊕E₈	E₅⊕E₉⊕E₁₃	A⊕E₄⊕E₈⊕E₁₂	More
E₁₄*	A⊕E₂	E₁₂⊕E₁₄	A⊕E₂⊕E₄	E₁₀⊕E₁₂⊕E₁₄	A⊕E₂⊕E₄⊕E₆	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 C₃₀**
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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