Symmetry operations are geometrical operations that interchange or transpose the atomic centers of a given molecule such that the newly obtained orientation of the molecule is physically indistuingishable to the original orientation. The following operations should be considered for three dimensional molecular systems:

Symmetry operations

Symbol	Operation
E	identity operation (do nothing)
σ	reflection through a mirror plane (σv, σh)
Cn	rotation around n-fold axis
i	inversion through a center of symmetry
Sn	rotation around n-fold axis + reflection in a plane perpendicular to axis of rotation

Each of these symmetry operations is connected to a symmetry element such as a plane of symmetry, a symmetry axis, a point of inversion, or a rotation-reflection (or improper) symmetry axis. These geometrical entities are denoted with the same symbols as the corresponding symmetry operations, but lack the operator "hat".
A symmetry axis is denoted as C_n if clockwise rotation by 360/n degrees leads to an unchanged configuration as described above. The subscript n is called the order of the axis. Multiple applications of the C_n operation are denoted with an additional superscript. Twofold rotation around the C₃ symmetry axis in ammonia (NH₃) is therefore described as C₃². In case there is more than one symmetry axis, the highest order axis is referred to as the principal axis.
The description of symmetry planes is modified in the presence of a symmetry axis. The symbol σ_v describes a symmetry plane vertical to the principal axis, while a symmetry plane running horizontally through the principal axis is denoted as σ_h. In highly symmetric systems one additional type of symmetry plane occurs that divides the angle between two C_n axes. This type of plane is denoted σ_d.


The complete symmetry properties of molecular systems are described through (symmetry) point groups, here using the Schoenflies symbols.

Point groups (Schoenflies)

Symbol	Symmetry operations	Abelian?
C1	E	yes
Cs	E, σ	yes
Ci	E, i	yes
Cn	E, Cn	yes
Sn	E, Sn (rotary-reflection)	yes
Cnv	E, Cn, n*σv	only C2v
Cnh	E, Cn, σh	yes
Dn	E, Cn, n*C2(vertical)	only D2
Dnh	E, Cn, n*C2(vertical), σh	only D2h
Dnd	E, Cn, n*C2(vertical), n*σv(to C2 axes)	no
Td	E, 6*σd, 4*C3, 3*S4	no
Oh	E, 6*σd, 3*σh, 6*C2, 4*C3, 3*C4, 3*S4, 4*S6,	no

Point groups are called Abelian if all their symmetry elements commute, that is, if application of the symmetry elements in one or another sequence leads to the same final result.