Self-action by conjugation

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There is reference to "Self-action by conjugation" on GroupWiki. The following piece of code will generate the same table in Mathematica.

Here row element is $"g"$ and column element is $"h"$ and cell is filled with $"hgh^{-1}"$

CODE:

Module[{newlist = List[], 
  inlist = List[]}, {list = ele = GroupElements[SymmetricGroup[3]],
  For[i = 1, i <= Length[list], i++,
    For[k = 1, k <= Length[list], k++, 
      AppendTo[inlist, 
       PermutationProduct[list[[i]], list[[k]], InversePermutation[list[[i]]]]]]
     {AppendTo[newlist, inlist], inlist = List[]}]
   TableForm[newlist, TableHeadings -> {ele, ele}]}]

OUTPUT:

The elements in left most corner and top most row are the elements of group and rest of the cells are the interaction between them. $\left( \begin{array}{ccccccc} & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] \\ \text{Cycles}[\{\}] & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] \\ \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] \\ \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] \\ \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] \\ \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] \\ \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] & \text{Cycles}[\{\}] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 3 & 2 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{ccc} 1 & 2 & 3 \\ \end{array} \right)\right] & \text{Cycles}\left[\left( \begin{array}{cc} 1 & 3 \\ \end{array} \right)\right] \\ \end{array} \right)$