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I would like to permute the elements of the set $A$ using symmetric group. How may I be able to do that? For example, in Sage,

G = SymmetricGroup(3)
for i in G: print [j*i*j^(-1) for j in G]

gives all the conjugacy classes for $S_3$ group. How may I able to do same with SymmetricGroup[3] is Mathematica.

Furthermore, I would like find the elements of $S_3$ that leaves the set $A=\{1, 2\}$ invariant or leaves each element fixed. How may I be able to do that?

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I find the OP questions somewhat unclear. In any case, the answers can be found in the tutorials "Permutations" and "Permutation Groups".

I would like to permute the elements of the set A using symmetric group.

In[44]:= Permute[{1,2,3},SymmetricGroup[3]]

Out[44]= {{1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}}

The Mathematica equivalent of the Sage code:

G = SymmetricGroup(3)
for i in G: print [j*i*j^(-1) for j in G]

should be written with other operators/functions, otherwise j*i*j^(-1) will simplify to i. Here I use NonCommutativeMultiply and Inverse:

In[42]:= Permute[NonCommutativeMultiply[j, i, Inverse[j]], SymmetricGroup[3]]

Out[42]= {j ** i ** Inverse[j], j ** Inverse[j] ** i, 
 i ** j ** Inverse[j], i ** Inverse[j] ** j, Inverse[j] ** j ** i, 
 Inverse[j] ** i ** j}

Furthermore, I would like find the elements of $S_3$ that leaves the set $A=\{1,2\}$ invariant [...]

This is unclear to me because:

In[47]:= GroupStabilizer[SymmetricGroup[3], {1, 2}]

Out[47]= PermutationGroup[{}]

hence

In[36]:= Permute[{1,2,3}, GroupStabilizer[SymmetricGroup[3], {1, 2}]]

Out[36]= {{}}

May be the following examples are helpful illustrations.

Fixing the 3d element using $S_3$:

In[50]:= Permute[Range[3], GroupStabilizer[SymmetricGroup[3], {3}]]

Out[50]= {{1, 2, 3}, {2, 1, 3}}

Fixing the elements {1,2} using $S_4$:

In[49]:= Permute[Range[4], GroupStabilizer[SymmetricGroup[4], {1, 2}]]

Out[49]= {{1, 2, 3, 4}, {1, 2, 4, 3}}
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