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Point groups

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Symmetry operations

E = identity

Cn = n-fold rotation

Sn = n-fold rotation plus reflection through a plane perpendicular to the axis of rotation

i = inversion through a centre of symmetry

$\sigma$v = reflection through a mirror plane (called “vertical”) parallel to the principal axis

$\sigma$h = reflection through a mirror plane (called “horizontal”) perpendicular to the principal axis

$\sigma$d = reflection through a vertical mirror plane bisecting the angle between two C2 axes

Irreducible representations

A = one-dimensional irreducible representation with character +1 under the principal rotation

B = one-dimensional irreducible representation with character -1 under the principal rotation

E = two-dimensional irreducible representation

T = three-dimensional irreducible representation

Table of several point groups

Nonaxial groups C1 Cs Ci
Cn groups C2 C3 C4 C5 C6 C7 C8
Dn groups D2 D3 D4 D5 D6 D7 D8
Cnv groups C2v C3v C4v C5v C6v C7v C8v
Cnh groups C2h C3h C4h C5h C6h
Dnh groups D2h D3h D4h D5h D6h D7h D8h
Dnd groups D2d D3d D4d D5d D6d D7d D8d
Sn groups S2 S4 S6 S8 S10 S12
Cubic groups T Th Td O Oh I Ih
Linear groups C$\infty$v D$\infty$h

Different orientations

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