- a 3-fold axis, associated with two symmetry operations: (+120 rotation) and (-120 rotation).
- 3 vertical planes, , , and associated with tree mirror reflections.

Now we add to these five symmetry operations the identity operator and show that all six
symmetry operations joint a group. Particularly, it is easy to see that
, where
the identity operation can be considered as a "product" of the two rotation operators
operations and . It can also be seen that
and
. Following this procedure we can build the "multiplication
table" presented below.

According to the Table 1, the "product" of each two symmetry transformations from
six , , , , , and is equivalent to one of
these transformations. It is clearly seen that the third and the fourth conditions of the group
are also valid. Thus, these six operators build a group. This group is known as group.
The total number of operations in a group is called the **group order**. Therefore, the
order of is 6.

Let us consider the symmetry group of molecule. The symmetry elements are:

- a 2-fold axis, associated with the symmetry operation: .
- two vertical planes, , and associated with two mirror reflections.

Now we add to these four symmetry operations the identity operator and show that all six
symmetry operations joint a group. The "multiplication table" presented below.

According to the Table 2, the "product" of each two symmetry transformations from six , , , and is equivalent to one of these transformations. It is clearly seen that the third and the fourth conditions of the group are also valid. Thus, these four operators build a group. This group is known as group. The group order of is 4.

Each point group is characterized by each own multiplication table.