Groups can have more properties than just a multiplication table. The illustration in the previous subsection is not amenable to illustrating this; the groups are to small. However, if we again consider the equilateral triangle, we note that it has further symmetry operations. Namely, those associated with reflections through a vertical plane through each vertex. Let us give these reflection operations the symbols dv, a! v and d"v, reflection through planes through vertex a, b, c, respectively. We may describe the operation by the transformations of the vertices a, b, c. For example dv : (a, b, c) ^ (a, c, b). By adding these three reflections operations to the rotations, we form the larger group of symmetry operations of the equilateral triangle G3 = {E, C3, C2, dv, a’v, d"v}. The multiplication table of the new group is given in Table 3. Another group with the same multiplication table as above can be constructed by considering the six permutations of the three letters a, b, c. Let E = (a, b, c), A = (c, a, b), B = (b, c, a), a’ = (a, c, b), b’ = (c, b, a), c’ = (b, a, c) and {E, A, B} are even permutations, {a’, b’, c’} are odd. This leads to the multiplication table 4, which is isomorphic to Table 3. The two multiplication tables illustrate the concept of a subgroup that is defined as: A set S of elements in group G is considered as a subgroup of G if:

1. all elements in S are also elements in G

2. for any two elements in S their product is in S

3. all elements in S satisfy the four group postulates.

From the presented multiplication tables we see that {E, C3, c3 } and {E, A, B} are subgroups; they are the only subgroups in their respective groups. These subgroups are the rotations in the former example, and the even permutations in the latter. While the remaining elements are associated with reflections and odd permutations, respectively. Furthermore, we see that two reflections are equivalent to a rotation, and two odd permutations are equivalent to an even permutation. Thus the operations {C3, C2} and dv, d’v, d"v and similarly {A, B} and {a’, b’, c’} belong in some sense to different sets. This property is illustrated by taking the transform T-1QT of each element Q in G by all elements T in G. For group {E, C3, C2, dv, ff’v, d"v} we obtain the following table of the transforms T-1 QT:

We note that {E}, {C3, C3 }, and { dv, d’v, d"v } transform into themselves and are thereby called classes. Classes play a leading role in the application of group theory to the solution of physical problems. In general physically significant properties can be associated with each class. In the solution of boundary value problems, different subspaces of the solution function space are assigned to each class.