Point Group S28

Point Group S₂₈

ε=exp(2πi/28)
S₂₈	E	S₂₈	C₁₄	(S₂₈)³	C₇	(S₂₈)⁵	(C₁₄)³	S₄	(C₇)²	(S₂₈)⁹	(C₁₄)⁵	(S₂₈)¹¹	(C₇)³	(S₂₈)¹³	C₂	(S₂₈)¹⁵	(C₇)⁴	(S₂₈)¹⁷	(C₁₄)⁹	(S₂₈)¹⁹	(C₇)⁵	(S₄)³	(C₁₄)¹¹	(S₂₈)²³	(C₇)⁶	(S₂₈)²⁵	(C₁₄)¹³	(S₂₈)²⁷
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
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
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*	i -i	-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*	-i i	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*	-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*	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*	-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*
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*	-i i	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	-ε^6* -ε^6*	-ε^3* -ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	i -i	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-ε^5* -ε^5*	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^4* ε^4*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*	1 1	ε^4* ε^4*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	ε^4* ε^4*
E₅*	1 1	ε^5* ε^5*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^6* -ε^6*	ε^3* ε^3*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	-ε^3* -ε^3*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	-ε^5* -ε^5*	-1 -1	-ε^5* -ε^5*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	-ε^3* -ε^3*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	ε^3* ε^3*	-ε^6* -ε^6*	-ε^* -ε^*	-ε^4* -ε^4*	ε^5* ε^5*
E₆*	1 1	ε^6* ε^6*	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	-ε^6* -ε^6*	-1 -1	-ε^6* -ε^6*	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^6* ε^6*	1 1	ε^6* ε^6*	-ε^2* -ε^2*	-ε^4* -ε^4*	ε^4* ε^4*	ε^2* ε^2*	-ε^6* -ε^6*	-1 -1	-ε^6* -ε^6*	ε^2* ε^2*	ε^4* ε^4*	-ε^4* -ε^4*	-ε^2* -ε^2*	ε^6* ε^6*
E₇*	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i	1 1	i -i	-1 -1	-i i
E₈*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*	1 1	-ε^6* -ε^6*	-ε^2* -ε^2*	ε^4* ε^4*	ε^4* ε^4*	-ε^2* -ε^2*	-ε^6* -ε^6*
E₉*	1 1	-ε^5* -ε^5*	-ε^4* -ε^4*	ε^* ε^*	-ε^6* -ε^6*	-ε^3* -ε^3*	ε^2* ε^2*	i -i	-ε^2* -ε^2*	ε^3* ε^3*	ε^6* ε^6*	-ε^* -ε^*	ε^4* ε^4*	ε^5* ε^5*	-1 -1	ε^5* ε^5*	ε^4* ε^4*	-ε^* -ε^*	ε^6* ε^6*	ε^3* ε^3*	-ε^2* -ε^2*	-i i	ε^2* ε^2*	-ε^3* -ε^3*	-ε^6* -ε^6*	ε^* ε^*	-ε^4* -ε^4*	-ε^5* -ε^5*
E₁₀*	1 1	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^2* ε^2*	-ε^2* -ε^2*	ε^6* ε^6*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^6* ε^6*	-ε^2* -ε^2*	ε^2* ε^2*	-ε^6* -ε^6*	-ε^4* -ε^4*	1 1	-ε^4* -ε^4*	-ε^6* -ε^6*	ε^2* ε^2*	-ε^2* -ε^2*	ε^6* ε^6*	ε^4* ε^4*	-1 -1	ε^4* ε^4*	ε^6* ε^6*	-ε^2* -ε^2*	ε^2* ε^2*	-ε^6* -ε^6*	-ε^4* -ε^4*
E₁₁*	1 1	-ε^3* -ε^3*	ε^6* ε^6*	ε^5* ε^5*	-ε^2* -ε^2*	ε^* ε^*	-ε^4* -ε^4*	-i i	ε^4* ε^4*	-ε^* -ε^*	ε^2* ε^2*	-ε^5* -ε^5*	-ε^6* -ε^6*	ε^3* ε^3*	-1 -1	ε^3* ε^3*	-ε^6* -ε^6*	-ε^5* -ε^5*	ε^2* ε^2*	-ε^* -ε^*	ε^4* ε^4*	i -i	-ε^4* -ε^4*	ε^* ε^*	-ε^2* -ε^2*	ε^5* ε^5*	ε^6* ε^6*	-ε^3* -ε^3*
E₁₂*	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*	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*	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*	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*
E₁₃*	1 1	-ε^* -ε^*	ε^2* ε^2*	-ε^3* -ε^3*	ε^4* ε^4*	-ε^5* -ε^5*	ε^6* ε^6*	i -i	-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*	-i i	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 = 28
Number of classes, irreps	n = 28
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	S₂₈	C₁₄	(S₂₈)³	C₇	(S₂₈)⁵	(C₁₄)³	S₄	(C₇)²	(S₂₈)⁹	(C₁₄)⁵	(S₂₈)¹¹	(C₇)³	(S₂₈)¹³	C₂	(S₂₈)¹⁵	(C₇)⁴	(S₂₈)¹⁷	(C₁₄)⁹	(S₂₈)¹⁹	(C₇)⁵	(S₄)³	(C₁₄)¹¹	(S₂₈)²³	(C₇)⁶	(S₂₈)²⁵	(C₁₄)¹³	(S₂₈)²⁷

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

**Binary products**
⊙	A	B	E₁*	E₂*	E₃*	E₄*	E₅*	E₆*	E₇*	E₈*	E₉*	E₁₀*	E₁₁*	E₁₂*	E₁₃*
A	A
B	B	A
E₁*	E₁	E₁₃	2A⊕E₂
E₂*	E₂	E₁₂	E₁⊕E₃	2A⊕E₄
E₃*	E₃	E₁₁	E₂⊕E₄	E₁⊕E₅	2A⊕E₆
E₄*	E₄	E₁₀	E₃⊕E₅	E₂⊕E₆	E₁⊕E₇	2A⊕E₈
E₅*	E₅	E₉	E₄⊕E₆	E₃⊕E₇	E₂⊕E₈	E₁⊕E₉	2A⊕E₁₀
E₆*	E₆	E₈	E₅⊕E₇	E₄⊕E₈	E₃⊕E₉	E₂⊕E₁₀	E₁⊕E₁₁	2A⊕E₁₂
E₇*	E₇	E₇	E₆⊕E₈	E₅⊕E₉	E₄⊕E₁₀	E₃⊕E₁₁	E₂⊕E₁₂	E₁⊕E₁₃	2A⊕2B
E₈*	E₈	E₆	E₇⊕E₉	E₆⊕E₁₀	E₅⊕E₁₁	E₄⊕E₁₂	E₃⊕E₁₃	2B⊕E₂	E₁⊕E₁₃	2A⊕E₁₂
E₉*	E₉	E₅	E₈⊕E₁₀	E₇⊕E₁₁	E₆⊕E₁₂	E₅⊕E₁₃	2B⊕E₄	E₃⊕E₁₃	E₂⊕E₁₂	E₁⊕E₁₁	2A⊕E₁₀
E₁₀*	E₁₀	E₄	E₉⊕E₁₁	E₈⊕E₁₂	E₇⊕E₁₃	2B⊕E₆	E₅⊕E₁₃	E₄⊕E₁₂	E₃⊕E₁₁	E₂⊕E₁₀	E₁⊕E₉	2A⊕E₈
E₁₁*	E₁₁	E₃	E₁₀⊕E₁₂	E₉⊕E₁₃	2B⊕E₈	E₇⊕E₁₃	E₆⊕E₁₂	E₅⊕E₁₁	E₄⊕E₁₀	E₃⊕E₉	E₂⊕E₈	E₁⊕E₇	2A⊕E₆
E₁₂*	E₁₂	E₂	E₁₁⊕E₁₃	2B⊕E₁₀	E₉⊕E₁₃	E₈⊕E₁₂	E₇⊕E₁₁	E₆⊕E₁₀	E₅⊕E₉	E₄⊕E₈	E₃⊕E₇	E₂⊕E₆	E₁⊕E₅	2A⊕E₄
E₁₃*	E₁₃	E₁	2B⊕E₁₂	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	E₃⊕E₉⊕E₁₃	A⊕E₆⊕E₁₀⊕E₁₂	More
E₄*	A⊕E₈	E₄⊕E₁₂	A⊕E₈⊕E₁₂	E₄⊕E₈⊕E₁₂	A⊕E₄⊕E₈⊕E₁₂	More
E₅*	A⊕E₁₀	E₅⊕E₁₃	A⊕E₈⊕E₁₀	E₃⊕E₅⊕E₁₃	A⊕E₂⊕E₈⊕E₁₀	More
E₆*	A⊕E₁₂	E₆⊕E₁₀	A⊕E₄⊕E₁₂	E₂⊕E₆⊕E₁₀	A⊕E₄⊕E₈⊕E₁₂	More
E₇*	A⊕2B	2E₇	3A⊕2B	3E₇	3A⊕4B	More
E₈*	A⊕E₁₂	E₄⊕E₈	A⊕E₄⊕E₁₂	E₄⊕E₈⊕E₁₂	A⊕E₄⊕E₈⊕E₁₂	More
E₉*	A⊕E₁₀	E₁⊕E₉	A⊕E₈⊕E₁₀	E₁⊕E₉⊕E₁₁	A⊕E₂⊕E₈⊕E₁₀	More
E₁₀*	A⊕E₈	E₂⊕E₁₀	A⊕E₈⊕E₁₂	E₂⊕E₆⊕E₁₀	A⊕E₄⊕E₈⊕E₁₂	More
E₁₁*	A⊕E₆	E₅⊕E₁₁	A⊕E₆⊕E₁₂	E₁⊕E₅⊕E₁₁	A⊕E₆⊕E₁₀⊕E₁₂	More
E₁₂*	A⊕E₄	E₈⊕E₁₂	A⊕E₄⊕E₈	E₄⊕E₈⊕E₁₂	A⊕E₄⊕E₈⊕E₁₂	More
E₁₃*	A⊕E₂	E₁₁⊕E₁₃	A⊕E₂⊕E₄	E₉⊕E₁₁⊕E₁₃	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	B⊕E₁	3	B⊕E₁
d (l=2)	5	Quadrupole	A⊕E₂⊕E₁₃	6	2A⊕E₂⊕E₁₃
f (l=3)	7	Octupole	B⊕E₁⊕E₃⊕E₁₂	10	2B⊕2E₁⊕E₃⊕E₁₂
g (l=4)	9	Hexadecapole	A⊕E₂⊕E₄⊕E₁₁⊕E₁₃	15	3A⊕2E₂⊕E₄⊕E₁₁⊕2E₁₃
h (l=5)	11	Dotricontapole	B⊕E₁⊕E₃⊕E₅⊕E₁₀⊕E₁₂	21	3B⊕3E₁⊕2E₃⊕E₅⊕E₁₀⊕2E₁₂
i (l=6)	13	Tetrahexacontapole	A⊕E₂⊕E₄⊕E₆⊕E₉⊕E₁₁⊕E₁₃	28	4A⊕3E₂⊕2E₄⊕E₆⊕E₉⊕2E₁₁⊕3E₁₃
j (l=7)	15	Octacosahectapole	B⊕E₁⊕E₃⊕E₅⊕E₇⊕E₈⊕E₁₀⊕E₁₂	36	4B⊕4E₁⊕3E₃⊕2E₅⊕E₇⊕E₈⊕2E₁₀⊕3E₁₂
k (l=8)	17	256-pole	A⊕E₂⊕E₄⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₁⊕E₁₃	45	5A⊕4E₂⊕3E₄⊕2E₆⊕E₇⊕E₈⊕2E₉⊕3E₁₁⊕4E₁₃
l (l=9)	19	512-pole	B⊕E₁⊕E₃⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀⊕E₁₂	55	5B⊕5E₁⊕4E₃⊕3E₅⊕E₆⊕2E₇⊕2E₈⊕E₉⊕3E₁₀⊕4E₁₂
m (l=10)	21	1024-pole	A⊕E₂⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀⊕E₁₁⊕E₁₃	66	6A⊕5E₂⊕4E₄⊕E₅⊕3E₆⊕2E₇⊕2E₈⊕3E₉⊕E₁₀⊕4E₁₁⊕5E₁₃
n (l=11)	23	2048-pole	B⊕E₁⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀⊕E₁₁⊕E₁₂	78	6B⊕6E₁⊕5E₃⊕E₄⊕4E₅⊕2E₆⊕3E₇⊕3E₈⊕2E₉⊕4E₁₀⊕E₁₁⊕5E₁₂
o (l=12)	25	4096-pole	A⊕E₂⊕E₃⊕E₄⊕E₅⊕E₆⊕E₇⊕E₈⊕E₉⊕E₁₀⊕E₁₁⊕E₁₂⊕E₁₃	91	7A⊕6E₂⊕E₃⊕5E₄⊕2E₅⊕4E₆⊕3E₇⊕3E₈⊕4E₉⊕2E₁₀⊕5E₁₁⊕E₁₂⊕6E₁₃
More

First nonvanshing multipole: Quadrupole

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 S₂₈**
L	2L+1	Term Splitting
S (L=0)	1	A
P (L=1)	3	A⊕E₁₃
D (L=2)	5	A⊕E₂⊕E₁₃
F (L=3)	7	A⊕E₂⊕E₁₁⊕E₁₃
G (L=4)	9	A⊕E₂⊕E₄⊕E₁₁⊕E₁₃
H (L=5)	11	A⊕E₂⊕E₄⊕E₉⊕E₁₁⊕E₁₃
I (L=6)	13	A⊕E₂⊕E₄⊕E₆⊕E₉⊕E₁₁⊕E₁₃

Last update November, 13^th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement