Check [11] at https://doi.org/10.31219/osf.io/r3jvu or here.

$S_2=\{(1),(12)\}$, let $a:=(1)$ and $b:=(12)$

$$ \begin{array}{c|cc} S_2 & a & b \\\hline
a & a & b \\ b & b & a \end{array} $$

$S_3=\{(1),(12),(13),(23),(123),(132)\}$

$a:=(1)$, $b:=(12)$, $c:=(13)$, $d:=(23)$, $e:=(123)$, $f:=(132)$

$$ \begin{array}{c|cc} S_3 & a & b & c & d & e & f \\\hline
a & a & b & c & d & e & f \\ b & b & a & e & f & c & d \\ c & c & f & a & e & d & b \\ d & d & e & f & a & b & c \\ e & e & d & b & c & f & a \\ f & f & c & d & b & a & e \end{array} $$