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D5h point group
not Abelian, 8(12) irreducible representations
Subgroups: Cs, C2, C5, D5, C2v, C5v, C5h

Character table
E 2C5 2(C5)2 5C'2 σh 2S5 2(S5)3 v
linear,
rotations
quadratic
A'1 1 1 1 1 1 1 1 1 x2+y2, z2
A'2 1 1 1 -1 1 1 1 -1 Rz
E'1 2 2cos(2π/5) 2cos(4π/5) 0 2 2cos(2π/5) 2cos(4π/5) 0 (x, y)
E'2 2 2cos(4π/5) 2cos(2π/5) 0 2 2cos(4π/5) 2cos(2π/5) 0 (x2-y2, xy)
A''1 1 1 1 1 -1 -1 -1 -1
A''2 1 1 1 -1 -1 -1 -1 1 z
E''1 2 2cos(2π/5) 2cos(4π/5) 0 -2 -2cos(2π/5) -2cos(4π/5) 0 (Rx, Ry) (xz, yz)
E''2 2 2cos(4π/5) 2cos(2π/5) 0 -2 -2cos(4π/5) -2cos(2π/5) 0

Product table
 A'1A'2E'1E'2A''1A''2E''1E''2
A'1A'1A'2E'1E'2A''1A''2E''1E''2
A'2A'2A'1E'1E'2A''2A''1E''1E''2
E'1E'1E'1A'1+A'2+E'2E'1+E'2E''1E''1A''1+A''2+E''2E''1+E''2
E'2E'2E'2E'1+E'2A'1+A'2+E'1E''2E''2E''1+E''2A''1+A''2+E''1
A''1A''1A''2E''1E''2A'1A'2E'1E'2
A''2A''2A''1E''1E''2A'2A'1E'1E'2
E''1E''1E''1A''1+A''2+E''2E''1+E''2E'1E'1A'1+A'2+E'2E'1+E'2
E''2E''2E''2E''1+E''2A''1+A''2+E''1E'2E'2E'1+E'2A'1+A'2+E'1

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