I hope it is ok to advertize some GAP code on this site.

Let G be a finite group and let

$R_*: \cdots \rightarrow R_4 \rightarrow R_3 \rightarrow R_2 \rightarrow R_1 \rightarrow R_0$

be a free $\mathbb ZG$-resolution of $\mathbb Z$.

A 3-cocycle with coefficients in $U(1)$ is a $\mathbb ZG$-linear homomorphism $f\colon R_3 \rightarrow U(1)$ such that the composite $R_4 \rightarrow R_3 \rightarrow U(1)$ is trivial. Using the Universal Coefficient Theorem I think we can represent such a $\mathbb ZG$-linear homomorphism by a $\mathbb ZG$-linear homomorphism $ f\colon R_3 \rightarrow \mathbb Z/m \mathbb Z$ where $m$ is the exponent of the third homology $H_3(G,\mathbb  Z$).

Let's call the cocycle $f\colon R_3 \rightarrow \mathbb Z/m$ a "standard cocycle" in the case where the resolution $R_*$ is the standard bar resolution. A standard cocycle can then be thought of as a function $F\colon G\times G\times G\rightarrow \mathbb Z/m \mathbb Z$ .

If we have any small resolution $R_*$, endowed with a contracting homotopy, then we can (in principle) construct a standard cocycle for each cohomology class. This can be done in GAP for many groups and provides examples of explicit cocycles which can be analyzed.

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EXAMPLE

As an example, let's construct a random cocycle $f\colon R_3 \rightarrow \mathbb Z/m\mathbb Z$ for the symmetric group $G=S_5$ with coefficients in $U(1)$. We use a small resolution $R$ constructed by GAP.

gap> G:=SymmetricGroup(5);;

gap> m:=Lcm(GroupHomology(G,3));

12

gap> R:=ResolutionFiniteGroup(G,4);;

gap> M:=CocycleCondition(R,3);;

gap> CocycleBasisMod4:=NullspaceModQ(TransposedMat(M),4);;

gap> CocycleBasisMod3:=NullspaceModQ(TransposedMat(M),3);;

gap> f:=(Random(CocycleBasisMod4) + Random(CocycleBasisMod3)) mod 12;

[ 2, 2, 2, 2, 1, 1, 3, 2, 1, 2, 2, 3, 2, 3, 1, 3, 0, 2, 4, 0 ]

Now let's convert f to a standard cocycle $F\colon G\times G\times G \rightarrow \mathbb Z/12\mathbb Z.

gap> F:=StandardCocycle(R,f,3,12);


And now let's evaluate $F(g,h,k)$ for, say, three random elements $g,h,k \in S_5$.

gap> g:=Random(G); h:=Random(G); k:=Random(G);

(2,4)(3,5)

(1,3)(2,4,5)

(1,5,2,4)

gap> F(g,h,k);

7

END OF EXAMPLE