TITLE: Explicit 3-cocycles for the symmetric group $S_6$
QUESTION [8 upvotes]: Hello,
this is a request for literature/a reference. I'm looking to do some calculations with the
symmetric group ($S_6$ and higher) and would be interested in explicit expressions for
3-cocycles, i.e. elements of $H^3(S_6, U(1))$.
Does anyone know whether these have already been calculated somewhere?

REPLY [2 votes]: I think Graham's answer is relevant to the question. 
A 3-cocycle on $G$ with values in an abelian group $A$ my also be described as a morphism of chain complexes from the standard free resolution $F(G)$ of $G$ to the chain complex say $(A,3)$ which is $A$ in dimension 3 and zero elsewhere. But it may be more convenient to compute another free resolution $C$ of $G$ and describe a morphism $C \to (A,3)$. A standard 3-cocycle then comes from the description of a morphism of chain complexes $F(G) \to C$,  which "in principle" is standard homological algebra. 
But the issue remains of the form required for the answer, and for this one probably needs to know  the place of the question in a research project. Why is the question asked? 
There are other ways of representing elements of $H^3(G,A)$, for example by "crossed sequences"
$$0 \to A \to C_2 \to C_1 \to G \to 1$$ 
where $C_2 \to C_1$ is a crossed module. There are good examples where $G,A, C_2,C_1$ are finite.  But intriguingly, for the current question $A$ is a topological abelian group!