# Conjugacy Classes of the Dihedral Group, D4

Let $D_4 = \langle r, s : r^4 = s^2 = 1, (rs)^2 = 1 \rangle = \{ 1, r, r^2, r^3, s, sr, sr^2, sr^3 \}$ where $r$ denotes the counterclockwise rotation translation, and $s$ denotes the flip translation.

The multiplication table for $D_4$ is given below:

$1$ | $r$ | $r^2$ | $r^3$ | $s$ | $rs$ | $r^2s$ | $r^3s$ | |
---|---|---|---|---|---|---|---|---|

$1$ | $1$ | $r$ | $r^2$ | $r^3$ | $s$ | $rs$ | $r^2s$ | $r^3s$ |

$r$ | $r$ | $r^2$ | $r^3$ | $1$ | $rs$ | $r^2s$ | $r^3s$ | $s$ |

$r^2$ | $r^2$ | $r^3$ | $1$ | $r$ | $r^2s$ | $r^3s$ | $s$ | $rs$ |

$r^3$ | $r^3$ | $1$ | $r$ | $r^2$ | $r^3s$ | $s$ | $rs$ | $r^2s$ |

$s$ | $s$ | $r^3s$ | $r^2s$ | $rs$ | $1$ | $r^3$ | $r^2$ | $r$ |

$rs$ | $rs$ | $s$ | $r^3s$ | $r^2s$ | $r$ | $1$ | $r^3$ | $r^2$ |

$r^2s$ | $r^2s$ | $rs$ | $s$ | $r^3s$ | $r^2$ | $r$ | $1$ | $r^3$ |

$r^3s$ | $r^3s$ | $r^2s$ | $rs$ | $s$ | $r^3$ | $r^2$ | $r$ | $1$ |

From this table we immediately see that $Z(D_4) = \{ 1, r^2 \}$. Each element of $Z(D_4)$ has a trivial conjugacy class, i.e., $[1] = \{ 1 \}$ and $[r^2] = \{ r^2 \}$.

Consider the element $r$. We have that:

(1)Thus the conjugacy class of $r$ is $[r] = \{ r, r^3 \}$.

Now consider the element $s$. We have that:

(2)Thus the conjugacy class of $s$ is $[s] = \{ s, r^2s \}$. There are only two remaining elements in $D_4$ that have not been assigned a conjugacy class yet. They are $rs$ and $r^3s$. Since $rs, r^3s \not \in Z(D_4)$ we have that $rs$ and $r^3s$ have nontrivial conjugacy classes. So $[rs] = \{ rs, r^3s \}$.

We can partition $D_4$ into its conjugacy classes by:

(3)And The Class Equation of $D_4$ is:

(4)