Title: Cayley table of group

Let \(S_{3}\) be a symmetric group i.e.  \(S_{3}=\left\{ e,a,b,c,d,f\right\} ,\) where

\begin{align*}
&e=\left(
\begin{array}{lll}
1 & 2 & 3 \\
1 & 2 & 3
\end{array}
\right) ,\text{  }a=\left(
\begin{array}{lll}
1 & 2 & 3 \\
2 & 3 & 1
\end{array}
\right) ,\text{  }b=\left(
\begin{array}{lll}
1 & 2 & 3 \\
3 & 1 & 2
\end{array}%
\right) ,\\&c=\left(
\begin{array}{lll}
1 & 2 & 3 \\
1 & 3 & 2%
\end{array}%
\right) ,\text{ }d=\left(
\begin{array}{lll}
1 & 2 & 3 \\
2 & 1 & 3%
\end{array}%
\right) ,\text{ }f=\left(
\begin{array}{lll}
1 & 2 & 3 \\
3 & 2 & 1%
\end{array}%
\right) .
\end{align*}

Multiplication of elements of \(S_{3}\) is given by \(\left( \color{green}{s} \circ \color{orange}{r} \right) \left( x\right) =\color{green}{s} \left( \color{orange}{r} \left( x\right) \right) .\)

NB! the first factor is \(\color{orange}{r}\), the second is \(\color{green}{s}\).

Problem.  Fill the Cayley table (the table of multiplication) of  \(S_3\), writting in the table e, a, b, c, d, f.

NB! In the table the first factor is orange, the second is green.

 | e | a | b | c | d | f
e | ___ | ___ | ___ | ___ | ___ | ___
a | ___ | ___ | ___ | ___ | ___ | ___
b | ___ | ___ | ___ | ___ | ___ | ___
c | ___ | ___ | ___ | ___ | ___ | ___
d | ___ | ___ | ___ | ___ | ___ | ___
f | ___ | ___ | ___ | ___ | ___ | ___

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