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.
The Conjugacy Classes are ,
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.
To find the irreducible representation, note that there are three Conjugacy Classes. Rule 5
requires that there be three irreducible representations satisfying
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 .
Using Group rule 1, we see that
Since there are only three Conjugacy Classes, this table is conventionally written simply as
Writing the irreducible representations in matrix form then yields
See also Dihedral Group, Finite Group D4, Finite Group Z6
© 1996-9 Eric W. Weisstein