1. Chapter 1
Point Group Symmetry

2. Symmetry Elements
E: Identity operation
Cn: Proper rotation

3. Symmetry Elements i: Inversion sh: Horizontal Mirror Plane
sv: Vertical Mirror Plane

4. Symmetry Elements Sn: Improper rotation: combination Cn and sh
S2 is equivalent to inversion (i)

5. Symmetry Groups Groups with no proper rotation axis
C1: Only E (i.e. no symmetry elements)
Cs: E and s
Ci: E and i
Sn: E, Sn (S1 = Cs; S2 = Ci)
Groups with one proper rotation axis
Cn: E, Cn only
Cnv: E, Cn, and n sv (linear unsymmetrical molecules are C∞v)
Cnh: E, Cn, and sh
Dihedral Groups: Groups with n C2 axes  to Cn
Dn: E, Cn, and n C2 axes  to Cn
Dnh: E, Cn, n C2 axes, and sh (linear symmetrical molecules are D∞h)
Dnd: E, Cn, n C2 axes, and n sv
Cubic Groups: Groups with more than one Cn (n ≥ 3)
Td: symmetry of a regular tetrahedron: 4 C3
Oh: symmetry of a regular octagon: 6 C4
Ih: symmetry of a regular icosahedron: 12 C5

6. Symmetry Decision Tree
Linear?
Find
principal
axes
C∞v or
D ∞h
More than one
Cn (n ≥ 3)
Cubic
T, O, I
Cs, Ci or C1
Cn is the principal axis?
nC2  to Cn?
S2n colinear w/ Cn?
n vertical mirror planes
S2n
sh?
sv?
Dnd
Dnh Dn Cn
Cnv
Cnh
Yes No
None
Physical Chemistry, Joseph H. Noggle, 2nd ed., Scott Foresman & Co, Glenview, IL, 1996, pg 840.