Symmetric Group on 4 Letters/Cayley Table

Cayley Table of Symmetric Group on $4$ Letters
The Cayley table of the symmetric group on $4$ letters can be written:
 * $\begin{array}{c|cccccc|cccccc|cccccc|cccccc}

\circ & e & t_{12} & t_{23} & r_{132} & r_{123} & t_{13} & t_{34} & v_a & r_{243} & f_{1432} & f_{1243} & r_{143} & r_{234} & f_{1342} & t_{24} & r_{142} & v_b & f_{1423} & f_{1234} & r_{134} & r_{124} & t_{14} & f_{1324} & v_c \\ \hline e &  e &  t_{12} &  t_{23} &  r_{132} &  r_{123} &  t_{13} &   t_{34} &  v_a &  r_{243} &  f_{1432} & f_{1243} & r_{143} &  r_{234} & f_{1342} & t_{24} & r_{142} & v_b & f_{1423} &  f_{1234} & r_{134} & r_{124} & t_{14} & f_{1324} & v_c \\ t_{12} &  t_{12} &  e &  r_{132} &  t_{23} &  t_{13} &  r_{123} &   v_a &  t_{34} &  f_{1432} &  r_{243} & r_{143} & f_{1243} &  f_{1342} & r_{234} & r_{142} & t_{24} & f_{1423} & v_b &  r_{134} & f_{1234} & t_{14} & r_{124} & v_c & f_{1324} \\ t_{23} &  t_{23} &  r_{123} &  e &  t_{13} &  t_{12} &  r_{132} &   r_{243} & f_{1243} &  t_{34} & r_{143} &  v_a &  f_{1432} &  t_{24} & v_b & r_{234} & f_{1423} & f_{1342} & r_{142} &  r_{124} & f_{1324} & f_{1234} & v_c & r_{134} & t_{14} \\ r_{132} &  r_{132} &  t_{13} &  t_{12} &  r_{123} &  e &  t_{23} &   f_{1432} & r_{143} &  v_a & f_{1243} &  t_{34} &  r_{243} &  r_{142} & f_{1423} & f_{1342} & v_b & r_{234} & t_{24} &  t_{14} & v_c & r_{134} & f_{1324} & f_{1234} & r_{124} \\ r_{123} &  r_{123} &  t_{23} &  t_{13} &  e &  r_{132} &  t_{12} &  f_{1243} &  r_{243} & r_{143} &  t_{34} &  f_{1432} &  v_a &  v_b & t_{24} & f_{1423} & r_{234} & r_{142} & f_{1342} &  f_{1324} & r_{124} & v_c & f_{1234} & t_{14} & r_{134} \\ t_{13} &  t_{13} &  r_{132} &  r_{123} &  t_{12} &  t_{23} &  e &  r_{143} &  f_{1432} & f_{1243} &  v_a &  r_{243} &  t_{34} &  f_{1423} & r_{142} & v_b & f_{1342} & t_{24} & r_{234} &  v_c & t_{14} & f_{1324} & r_{134} & r_{124} & f_{1234} \\ \hline t_{34} &  t_{34} &  v_a & r_{234} & f_{1342} & f_{1234} & r_{134} &   e &  t_{12} & t_{24} & r_{142} & r_{124} & t_{14} &   t_{23} &  r_{132} &  r_{243} &  f_{1432} & f_{1324} & v_c &   r_{123} &  t_{13} & f_{1243} & r_{143} & v_b & f_{1423} \\ v_a &  v_a &  t_{34} & f_{1342} & r_{234} & r_{134} & f_{1234} &   t_{12} &  e & r_{142} & t_{24} & t_{14} & r_{124} &   r_{132} &  t_{23} &  f_{1432} &  r_{243} & v_c & f_{1324} &   t_{13} &  r_{123} & r_{143} & f_{1243} & f_{1423} & v_b \\ r_{243} &  r_{243} & f_{1243} & t_{24} & v_b & r_{124} & f_{1324} &   t_{23} &  r_{123} & r_{234} & f_{1423} & f_{1234} & v_c &   e &  t_{13} &  t_{34} & r_{143} & r_{134} & t_{14} &   t_{12} &  r_{132} &  v_a &  f_{1432} & f_{1342} & r_{142} \\ f_{1432} &  f_{1432} & r_{143} & r_{142} & f_{1423} & t_{14} & v_c &   r_{132} &  t_{13} & f_{1342} & v_b & r_{134} & f_{1324} &   t_{12} &  r_{123} &  v_a & f_{1243} & f_{1234} & r_{124} &   e &  t_{23} &  t_{34} &  r_{243} & r_{234} & t_{24} \\ f_{1243} & f_{1243} & r_{243} & v_b & t_{24} & f_{1324} & r_{124} &   r_{123} &  t_{23} & f_{1423} & r_{234} & v_c & f_{1234} &   t_{13} &  e & r_{143} &  t_{34} & t_{14} & r_{134} &   r_{132} &  t_{12} &  f_{1432} &  v_a & r_{142} & f_{1342} \\ r_{143} & r_{143} & f_{1432} & f_{1423} & r_{142} & v_c & t_{14} &   t_{13} &  r_{132} & v_b & f_{1342} & f_{1324} & r_{134} &   r_{123} &  t_{12} & f_{1243} &  v_a & r_{124} & f_{1234} &   t_{23} &  e &  r_{243} &  t_{34} & t_{24} & r_{234} \\ \hline r_{234} & r_{234} & f_{1234} & t_{34} & r_{134} &  v_a & f_{1342} &  t_{24} & r_{124} &  e & t_{14} &  t_{12} & r_{142} &   r_{243} & f_{1324} &  t_{23} & v_c &  r_{132} &  f_{1432} &  f_{1243} & v_b &  r_{123} & f_{1423} &  t_{13} & r_{143} \\ f_{1342} & f_{1342} & r_{134} & v_a & f_{1234} &  t_{34} & r_{234} &  r_{142} & t_{14} &  t_{12} & r_{124} &  e & t_{24} &   f_{1432} & v_c &  r_{132} & f_{1324} &  t_{23} &  r_{243} &  r_{143} & f_{1423} &  t_{13} & v_b &  r_{123} & f_{1243} \\ t_{24} & t_{24} & r_{124} & r_{243} & f_{1324} & f_{1243} & v_b &  r_{234} & f_{1234} &  t_{23} & v_c &  r_{123} & f_{1423} &   t_{34} & r_{134} &  e & t_{14} &  t_{13} & r_{143} &   v_a & f_{1342} &  t_{12} & r_{142} &  r_{132} &  f_{1432} \\ r_{142} & r_{142} & t_{14} & f_{1432} & v_c & r_{143} & f_{1423} & f_{1342} & r_{134} &  r_{132} & f_{1324} &  t_{13} & v_b &   v_a & f_{1234} &  t_{12} & r_{124} &  r_{123} & f_{1243} &   t_{34} & r_{234} &  e & t_{24} &  t_{23} &  r_{243} \\ v_b & v_b & f_{1324} & f_{1243} & r_{124} & r_{243} & t_{24} &  f_{1423} & v_c &  r_{123} & f_{1234} &  t_{23} & r_{234} &  r_{143} & t_{14} &  t_{13} & r_{134} &  e &  t_{34} &   f_{1432} & r_{142} &  r_{132} & f_{1342} &  t_{12} &  v_a \\ f_{1423} & f_{1423} & v_c & r_{143} & t_{14} & f_{1432} & r_{142} &  v_b & f_{1324} &  t_{13} & r_{134} &  r_{132} & f_{1342} &  f_{1243} & r_{124} &  r_{123} & f_{1234} &  t_{12} &  v_a &   r_{243} & t_{24} &  t_{23} & r_{234} &  e &  t_{34} \\ \hline f_{1234} & f_{1234} & r_{234} & r_{134} & t_{34} & f_{1342} &  v_a &  r_{124} & t_{24} & t_{14} &  e & r_{142} &  t_{12} &  f_{1324} &  r_{243} & v_c &  t_{23} &  f_{1432} &  r_{132} &  v_b & f_{1243} & f_{1423} &  r_{123} & r_{143} &  t_{13} \\ r_{134} & r_{134} & f_{1342} & f_{1234} & v_a & r_{234} &  t_{34} &  t_{14} & r_{142} & r_{124} &  t_{12} & t_{24} &  e &  v_c & f_{1432} & f_{1324} &  r_{132} &  r_{243} &  t_{23} &  f_{1423} & r_{143} & v_b &  t_{13} & f_{1243} &  r_{123} \\ r_{124} & r_{124} & t_{24} & f_{1324} & r_{243} & v_b & f_{1243} &  f_{1234} & r_{234} & v_c &  t_{23} & f_{1423} &  r_{123} &  r_{134} &  t_{34} & t_{14} &  e & r_{143} &  t_{13} &  f_{1342} &  v_a & r_{142} &  t_{12} &  f_{1432} &  r_{132} \\ t_{14} & t_{14} & r_{142} & v_c & f_{1432} & f_{1423} & r_{143} &  r_{134} & f_{1342} & f_{1324} &  r_{132} & v_b &  t_{13} &  f_{1234} &  v_a & r_{124} &  t_{12} & f_{1243} &  r_{123} &  r_{234} &  t_{34} & t_{24} &  e &  r_{243} &  t_{23} \\ f_{1324} & f_{1324} & v_b & r_{124} & f_{1243} & t_{24} & r_{243} &  v_c & f_{1423} & f_{1234} &  r_{123} & r_{234} &  t_{23} &  t_{14} & r_{143} & r_{134} &  t_{13} &  t_{34} &  e &  r_{142} &  f_{1432} & f_{1342} &  r_{132} &  v_a &  t_{12} \\ v_c & v_c & f_{1423} & t_{14} & r_{143} & r_{142} & f_{1432} &  f_{1324} & v_b & r_{134} &  t_{13} & f_{1342} &  r_{132} &  r_{124} & f_{1243} & f_{1234} &  r_{123} &  v_a &  t_{12} &  t_{24} &  r_{243} & r_{234} &  t_{23} &  t_{34} &  e \\ \end{array}$

where the expression for $S_4$ in cycle notation is given as: