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.
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