Evaluation of the commutants

commutator[a_,b_]:=PermutationProduct[InversePermutation[a],InversePermutation[b],a,b]
commutatorGroup[list_]:=Union[Flatten[Table[commutator[list[[i]],list[[j]]],{i,1,Length[list]},{j,1,Length[list]}]]]

Linear equation

monodromyGroup=GroupElements[SymmetricGroup[1]]

Quadratic equation

monodromyGroup=GroupElements[SymmetricGroup[2]]
commutatorGroup[monodromyGroup]

Cubic equation

monodromyGroup=GroupElements[SymmetricGroup[3]]
commutatorGroup[monodromyGroup]
commutatorGroup[commutatorGroup[monodromyGroup]]

Quartic equation

monodromyGroup=GroupElements[SymmetricGroup[4]]
commutatorGroup[monodromyGroup]
commutatorGroup[commutatorGroup[monodromyGroup]]
commutatorGroup[commutatorGroup[commutatorGroup[monodromyGroup]]]

Quintic equation

monodromyGroup=GroupElements[SymmetricGroup[5]]
commutatorGroup[monodromyGroup]
commutatorGroup[commutatorGroup[monodromyGroup]]

Length[monodromyGroup]
Length[commutatorGroup[monodromyGroup]]
Length[commutatorGroup[commutatorGroup[monodromyGroup]]]