Note that the exceptional low-dimensional homology of the alternating group (formula_62 corresponding to non-trivial abelianization, and formula_63 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 formula_64 extends to formula_65 and the triple covers of A6 and A7 extend to triple covers of S6 and S7 – but these are not "homological" – the map formula_66 does not change the abelianization of S4, and the triple covers do not correspond to homology either.