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) |

(15) |

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) |

(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) |

© 1996-9

1999-05-26