List of Character Tables For Chemically Important 3D Point Groups - Character Tables - Linear (cylindrical) Groups

Linear (cylindrical) Groups

These groups are characterized by having a proper rotation axis C∞ around which the symmetry is invariant to any rotation.

Point
Group		 Character Table
C∞v
E 2 C∞Φ ... ∞ σv
A1=Σ+ 1 1 ... 1 z x2 + y2, z2
A2=Σ− 1 1 ... −1 Rz
E1=Π 2 2 cos(Φ) ... 0 (x, y), (Rx, Ry) (xz, yz)
E2=Δ 2 2 cos(2Φ) ... 0 (x2 - y2, xy)
E3=Φ 2 2 cos(3Φ) ... 0
... ... ... ... ...
D∞h
E 2 C∞Φ ... ∞ σv i 2 S∞Φ ... ∞ C2
Σg+ 1 1 ... 1 1 1 ... 1 x2 + y2, z2
Σg− 1 1 ... −1 1 1 ... −1 Rz
Πg 2 2 cos(Φ) ... 0 2 −2 cos(Φ) .. 0 (Rx, Ry) (xz, yz)
Δg 2 2 cos(2Φ) ... 0 2 2 cos(2Φ) .. 0 (x2 − y2, xy)
... ... ... ... ... ... ... ... ...
Σu+ 1 1 ... 1 −1 −1 ... −1 z
Σu− 1 1 ... −1 −1 −1 ... 1
Πu 2 2 cos(Φ) ... 0 −2 2 cos(Φ) .. 0 (x, y)
Δu 2 2 cos(2Φ) ... 0 −2 −2 cos(2Φ) .. 0
... ... ... ... ... ... ... ... ...

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