Generating set of groups

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To see the meaning of generators of any group one must refer to wikipedia.
There is API in Mathematica to find the generators of any group. This blog is about how a group is generated from generating sets. This works fine for SymmetricGroup of order 3 but for order 4 it generates partial number of sets but definitely tells how it works

The case considered here is of Symmetric Group of order 3.

CODE:

Module[{newlist = List[], inlist = List[]},
 {ele = SymmetricGroup[3],
  list = GroupGenerators[ele],
  Print["Generators=", list],
  AppendTo[newlist, list];
  For[i = 1, i <= Length[list], i++,
   For[k = 1, k <= Length[list], k++,
     {
      Print[{list[[k]], list[[i]]},
       "=" PermutationProduct[list[[k]], list[[i]]]],
      AppendTo[inlist, {list[[k]], list[[i]]}],
      AppendTo[inlist, PermutationProduct[list[[k]], list[[i]]]]
   
      }]
    {AppendTo[newlist, inlist], inlist = List[]}],
  Print["NewList=", newlist];
  }]



OUTPUT:

For SymmetricGroup[3]

Generating set  = {{Cycles[{{1,2}}],Cycles[{{1,2,3}}]}
Generated group elements =
{{Cycles[{{1,2}}],Cycles[{{1,2}}]}=Cycles[{}],{Cycles[{{1,2,3}}],Cycles[{{1,2}}]}=Cycles[{{2,3}}]},{{Cycles[{{1,2}}],Cycles[{{1,2,3}}]}=Cycles[{{1,3}}],{Cycles[{{1,2,3}}],Cycles[{{1,2,3}}]}=Cycles[{{1,3,2}}]}}

For SymmetricGroup[4]


Generating set  = {{Cycles[{{1,2}}],Cycles[{{1,2,3,4}}]},
Generated group elements =
{{Cycles[{{1,2}}],Cycles[{{1,2}}]}=Cycles[{}],
{Cycles[{{1,2,3,4}}],Cycles[{{1,2}}]}=Cycles[{{2,3,4}}]},
{{Cycles[{{1,2}}],Cycles[{{1,2,3,4}}]}=Cycles[{{1,3,4}}],
{Cycles[{{1,2,3,4}}],Cycles[{{1,2,3,4}}]}=Cycles[{{1,3},{2,4}}]}}

There shall be 24 elements but here we see only 6, but in similar fashion we can generate other elements of sets from generating sets.