Let $G$ be a finite group and $A$ an abelian group.  Recall the cochain groups
$$
	C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
	\delta : C^k \to C^{k+1}
$$
$$
	(\delta f)(g_1, \ldots, g_{k+1}) = f(g_2, \ldots, g_{n+1}) + \cdots +
			(-1)^i f(g_1, \ldots, g_i g_{i+1}, \ldots, g_{n+1}) + \cdots +
			f(g_1, \ldots, g_n)
$$
for the group cohomology of $G$ with coefficients in $A$.

To each $k$-tuple $(g_1,\ldots, g_k)$ we can associate a labeling of the (oriented) edges of the standard $k$-simplex as follows.  The oriented edge from $i$ to $j$, with $i<j$, is labeled by $g_{i+1}g_{i+2}\cdots g_j$, and the edge from $j$ to $i$ is labeled by the inverse of the $i$-to-$j$ label.  The canonical action of the permutation group $S_{k+1}$ on the $k$-simplex leads to an action of $S_{k+1}$ on $G^k$.  Define an action of $S_{k+1}$ on $C_k$ by
$$
	(\sigma f)(g_1, \ldots, g_k) = (-1)^{|\sigma|} f(\sigma(g_1, \ldots, g_k)) .
$$
($|\sigma|$ denotes the parity of the permutation.)

One can show (unless I've made a foolish mistake) that any group cohomology class can be represented by a $k$-cochain which is invariant under the action of $S_{k+1}$.  (Sketch of proof: consider a more parsimonious model for the classifying space $BG$.)  So, for example, for 1-cochains we want
$$
	f(g) = - f(g^{-1})
$$
and for 2-cochains we want
$$
	f(g,h) = f(h, (gh)^{-1}) = f((gh)^{-1}, g) = -f(h^{-1}, g^{-1}) = -f(g^{-1}, gh) = -f(gh, h^{-1}) .
$$

> **My question:** Where can I find a reference for this fact about group cohomology?

(The reason I care:  "In the wild" one frequently comes across $k$-simplices with edges labeled as above, but these $k$-simplices don't come equipped with an isomorphism to the standard $k$-simplex.  To work with standard group cochains one would need to break the symmetry and choose such an isomorphism.  I'd like to avoid doing that.)