Note that the exceptional low-dimensional homology of the alternating group ( H 1 ( A 3 ) ≅ ≅ H 1 ( A 4 ) ≅ ≅ C 3 , corresponding to non-trivial abelianization, and H 2 ( A 6 ) ≅ ≅ H 2 ( A 7 ) ≅ ≅ C 6 , due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map A 4 ↠ ↠ C 3 extends to S 4 ↠ ↠ S 3 , and the triple covers of A 6 and A 7 extend to triple covers of S 6 and S 7 – but these are not homological – the map S 4 ↠ ↠ S 3 does not change the abelianization of S 4, and the triple covers do not correspond to homology either.