Dihedral Group D3/Cayley Table

Cayley Table of Dihedral Group $D_3$

The Cayley table of the dihedral group $D_3$ can be written:

$\begin{array}{c|cccccc}

     & e     & a     & a^2   & b     & a b   & a^2 b \\

\hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$

where the group presentation of $D_3$ is given as:

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$