I am trying to understand the Cayley graph for the group $D_3$, which from Mathematica, I got:
I tried to get the multiplication table in Mathematica:
$\left( \begin{array}{cccccc} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 1 & 4 & 3 & 6 & 5 \\ 3 & 5 & 1 & 6 & 2 & 4 \\ 4 & 6 & 2 & 5 & 1 & 3 \\ 5 & 3 & 6 & 1 & 4 & 2 \\ 6 & 4 & 5 & 2 & 3 & 1 \\ \end{array} \right)$
where the first row and column are the table headings.
How does this multiplication table relate to the graph?
Thanks!
The table and graph are related as follows. Let $X=\{1,2,3,4,5,6\}$ and let $S=\{2,4\}$. The table is missing its "headings" so that the actual table looks like this $$ \begin{array}{c|cccccc} & 1 & 2& 3& 4& 5&6\\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 2 & 1 & 4 & 3 & 6 & 5 \\ 3 & 3 & 5 & 1 & 6 & 2 & 4 \\ 4 & 4 & 6 & 2 & 5 & 1 & 3 \\ 5 & 5 & 3 & 6 & 1 & 4 & 2 \\ 6 & 6 & 4 & 5 & 2 & 3 & 1 \\ \end{array} $$ where $i\cdot j$ is equal to the number in the $i$th row and $j$th column (so for example $4\cdot 2 =6$ and $2\cdot 4=3$). Next notice that $S$ generates $X$ under the above multiplication. To see this note that $2^2=1,2,4,4^2=5,2\cdot 4=3,2\cdot 4^2=6$ are all the elements of $X$. Now let us turn to the graph, the vertices are the elements of $X$ and there is a red edge from $x$ to $y$ if $x\cdot 2 =y$. Similarly there is purple edge from $x$ to $y$ if $x\cdot 4=y$.