Nonaxial symmetriesThis class includes C1, Ci, and Cs, which have no proper or improper rotation axis.

C1 GroupC1 has only one symmetry operation, {E}. The order of C1 group is 1. Molecules in this group have no symmetry, which means we can not perform rotation, reflection of a mirror plane, etc. And the only symmetry operation is identity, E.

Ci GroupCi has 2 symmetry operations, {E, i}. The order of C i group is 2. Molecules in this group have low symmetry, an inversion center. For example,C2H2F2Cl2 has an inversion center.

Figure 2.2 C2H2F2Cl2. Point group is Ci.

.

Cs GroupCs has 2 symmetry operations, {E, σ}. The order of Cs group is 2. Molecules in this group have low symmetry, a mirror plane. For example,CH2BrCl has a mirror plane.

Figure 2.3 CH2BrCl. Point group is Cs

Cyclic symmetriesThis class includes Cn, Cnh, Cnv, and Sn, which have only one proper or improper rotation axis.

Cyclic group: Cn group Cn (nσ2)

symmetry elements, E and Cn.

And n symmetry operations, {E, Cn1, Cn

2, … , Cnn-1}

The order of Cn group is n.

Pyramidal group: Cnv group

For Cnv group, symmetry elements are E, Cn, and nσv.

And symmetry operations are {E, Cnk(k=1, … ,n-1), nσv }

The order of Cnv group is 2n. For example, NH3 has a C3 axis and three mirror planes σv.

Therefore, the point group of NH3 is C3v.

Now we can generate a group multiplication table for NH3:

Table 2.2 Group multiplication table of symmetry operation of NH3 molecule

This C3v group, as what is mentioned before, has all the properties of a group in mathmetics. And all the molecules that have one C3 axis and 3 mirror planes such as NH3 molecule can be assigned to this C3v group. In the same way, the operations in the following groups also have all the properties of a mathmetical group and can generate a multiplicaiton table.

Reflection group: Cnh group

For Cnh group, symmetry elements are E, Cn, σh, and Sn.

And symmetry operations are {E, Cnk(k=1, … ,n-1), σh, σh Cn

m(m=1, … ,n-1)}

The order of Cnh group is 2n.

For example, point group of C2H2F2 is C2h.

Improper rotation group: Sn group

If n=1, S1=Cs

If n=2, S2=Ci

If n=odd number, Sn (n=3, 5, 7 …) = Cnh

For example, operations in S3are the same as C3h, e.g. B(OH)3.

S3={E, S3, S32, S3

3, S34, S3

5} ={E, S3, C32, σh, C3, S3

5}= C3h

Figure 2.7 B(OH)3. Point group C3h.This picture is drawn by MacMolPlt.

Therefore, for Sn group, n can only be 4, 6, 8 …..

The symmetry elements are E and Sn. And symmetry operations are {E, Snk(k=1,

… ,n-1)}. The order of Sn group is n.

For example, the point group of 1,3,5,7 -tetrafluoracyclooctatetrane is S4.

Dihedral symmetriesThis class includes Dn, Dnh, and Dnd, which have one proper rotation Cn axis and n C2 axis perpendicular to Cn axis.

Dihedral group: Dn groupFor Dn group, symmetry elements are E, Cn, and nC2 (σCn).

And symmetry operations are {E, Cnk(k=1, … ,n-1), nC2}

The order of Dn group is 2n.

For example, the point group of [Co(en)3]3+ is D3.

Prismatic group: D nh group For Dnh group, symmetry elements are E, Cn, σh,and nC2 (σCn).

And symmetry operations are {E, Cnk(k=1, … ,n-1), ?h, Sn

m(m=1, … ,n-1), nC2, nσv}

The order of Dnh group is 4n.

For example, the point group of benzene is D6h.

Antiprismatic group: D nd group For Dnd group, symmetry elements are E, Cn, σd, and nC2 (σCn).

And symmetry operations are {E, Cnk(k=1, … ,n-1), S2n

m(m=1, … ,2n-1), nC2, nσd}

The order of Dnd group is 4n.

For example, pinot group of C2H6 is D3d.

Polyhedral symmetriesThis class includes T, Th, Td, O, Oh, I and Ih, which have more than two high-order axes.

Cubic groups: T, Th, Td, O, Oh

These groups do not have a C5 peoper rotation axis.

T group

For T group, symmetry elements are E, 4C3, and 3C2.

And symmetry operations are {E, 4C3, 4C32, 3C2}

The order of T group is 12.

Th group

For Td group, symmetry elements are E, 3C2, 4C3, i, 4S6 and 3σh.

And symmetry operations are {E, 4C3, 4C32, 3C2, i, 4S6, 4S6

5, 3σh}

The order of Td group is 24.

Td group

For Td group, symmetry elements are E, 3C2, 4C3, 3S4 and 6σd.

And symmetry operations are {E, 8C3, 3C2, 6S4, 6σd}

The order of Td group is 24.

For example, the point group of CCl4 is Td.

Point group is Td. The figure is drawn by ACD Labs 11.0.

O group

For O group, symmetry elements are E, 3C4, 4C3, and 6C2.

And symmetry operations are {E, 8C3, 3C2, 6C4, 6C2}

The order of O group is 24.

Oh group

For Oh group, symmetry elements are E, 3S4, 3C4, 6C2, 4S6, 4C3, 3?h, 6σd, and i.

And symmetry operations are {E, 8C3, 6C2, 6C4, 3C2, i, 6S4, 8S6, 3σh, 6σd}

The order of Oh group is 48.

For example, the point group of SF6 is Oh.

Icosahedral groups: I, Ih

These groups have a C5 peoper rotation axis.

I group For I group, symmetry elements are E, 6C5, 10C3, and 15C2.

And symmetry operations are {E, 15C5, 12C52, 20C3, 15C2}

The order of I group is 60.

I h group For Ih group, symmetry elements are E, 6S10, 10S6, 6C5, 10C3, 15C2 and 15σ.

And symmetry operations are {E, 15C5, 12C52, 20C3, 15C2, i, 12S10, 12S10

3, 20S6, 15σ}

The order of Ih group is 120.

For example, the point group of C60 is Ih.

Linear groupsThis class includes C?v and D?h, which are the symmetry of linear molecules.

C ∞v group For C∞v group, symmetry elements are E, C∞ and ∞σv.

such as CO, HCN, NO, HCl.

D ∞h group For Dσh group, symmetry elements are E, C∞ ∞σv , σh, i, and ∞C2.

such as CO2, O2, N2.