If you interchange two zeros nothing will happen :)
In[3]:= Permute[{1, 2, 3}, SymmetricGroup[3]]
Out[3]= {{1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}}
In[4]:= Permute[{1, 2, 3}, AlternatingGroup[3]]
Out[4]= {{1, 2, 3}, {2, 3, 1}, {3, 1, 2}}