## Finite Group D3

The Dihedral Group is one of the two groups of Order 6. It is the non-Abelian group of smallest Order. Examples of include the Point Groups known as , , , , the symmetry group of the Equilateral Triangle, and the group of permutation of three objects. Its elements satisfy , and four of its elements satisfy , where 1 is the Identity Element. The Cycle Graph is shown above, and the Multiplication Table is given below.

 1 1 1 1 1 1 1 1

The Conjugacy Classes are ,

 (1) (2) (3) (4) (5)

and , ,
 (6) (7)

A reducible 2-D representation using Real Matrices can be found by performing the spatial rotations corresponding to the symmetry elements of . Take the z-Axis along the axis.

 (8) (9) (10) (11) (12) (13)

To find the irreducible representation, note that there are three Conjugacy Classes. Rule 5 requires that there be three irreducible representations satisfying

 (14)

so it must be true that
 (15)

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

 1 1 1 1 1 1 1

To find a representation orthogonal to the totally symmetric representation, we must have three and three 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 .

 1 1 1 1 1 1 1 1 1 1

Using Group rule 1, we see that

 (16)

so the final representation for 1 has Character 2. Orthogonality with the first two representations (rule 3) then yields the following constraints:

 (17) (18)

Solving these simultaneous equations by adding and subtracting (18) from (17), we obtain , . The full Character Table is then

 1 1 1 1 1 1 1 1 1 1 2 0 0 0

Since there are only three Conjugacy Classes, this table is conventionally written simply as

 1 1 1 1 1 1 2 0

Writing the irreducible representations in matrix form then yields

 (19) (20) (21) (22) (23) (24)