This seems like a very nice way to obtain specific class representatives! I assume the fact that we get 96 classes instead of 48 comes from $|Ext(H_2(G,\mathbb{Z}),A)|=2$?

Thank you, I had a look at GAP/HAP already, but is there a way to find out which cohomology class the cocycle computed this way is in? It seems to me like you can never be sure to even get a nontrivial element.

$H^3(S_6,U(1))=Z_2\times Z_2 \times Z_{12}$, this I computed using GAP. I am looking for an explicit representative, i.e. a function $f:G\times G \times G \rightarrow U(1)$ satisfying the 3-cocycle condition (that is not a coboundary, obviously).