Dihedral Group D_3

The dihedral group is a particular instance of one of the two distinct abstract groups of group order 6. Unlike the cyclic group (which is an Abelian group), is a non-Abelian group. In fact, is the non-Abelian group having smallest group order.

Examples of include the point groups known as and , the symmetry group of the equilateral triangle (Arfken 1985, p. 246), and the symmetric group on three objects (Arfken 1985, p. 249).

The cycle graph of is shown above. has cycle index given by

Its multiplication table is illustrated above and enumerated below, where 1 denotes the identity element. Equivalent but slightly different forms are given by (Arfken 1985, p. 247) and Cotton (1990, p. 12), the latter of which denotes the abstract group of by .

An irreducible representation of in two dimensions using real matrices has generators and , where is a reflection in a line through the center and a vertex of an equilateral triangle, and is a rotation by about its center. The multiplication table above corresponds to the following matrices:

The elements , , , and of satisfy , the elements , , and satisfy , the elements , , , and satisfy , and all elements satisfy .

The conjugacy classes are , , and . There are 6 subgroups of : , , , , , and . Of these, the subgroups , , and are normal.

To find the irreducible representations, note that there are three conjugacy classes. The fifth rule of irreducible representations requires that there be three irreducible representations, and the second rule requires that

so it must be true that

By rule 6, we can let the first irreducible representation have all 1s.

To find the other one-dimensional irreducible representation, we must have three and three group characters. We can also add the constraint that the components of the identity element 1 be positive. The three conjugacy classes have 1, 2, and 3 elements. Since we need a total of three s and we have required that a occur for the conjugacy class of order 1, the remaining +1s must be used for the elements of the conjugacy class of order 2, i.e., and .

Using irreducible representation rule 1, we see that

so the final irreducible representation for 1 has group character 2. Orthogonality with the first two irreducible representations (rule 3) then yields the following constraints, where and are the unknown group characters on and :

Solving these simultaneous equations, we obtain , . The full character table is then

Since there are only three conjugacy classes, this table is conventionally written simply as

The direct sum of these three irreducible representations can be written in matrix form as