3 years later.

My recent paper (https://arxiv.org/abs/1703.03266) answers this question.

More precisely,
let $\mathbb{k}$ be an algebraically closed field of characteristic zero. By $\mathbb{k}^*$ we denote the multiplicative group $\mathbb{k}-\{0\}$. Let $G=\mathbb{Z}_{m_{1}}\times\cdots \times\mathbb{Z}_{m_{n}}$ where $m_i|m_{i+1}$ for $1\le i\le n-1$ and $(B_{\bullet},\partial_{\bullet})$ be its normalized bar resolution.
Applying $\text{Hom}_{\mathbb{Z}G}(-,\mathbb{k}^*)$ one gets a complex $(B^*_{\bullet},\partial^*_{\bullet})$.

Let $\alpha:=(\alpha_{11},\dots,\alpha_{1n},\dots,\alpha_{k1},\dots,\alpha_{kn})$ where $0\leq\alpha_{ij}<m_{j}$ for $1\leq i\leq k$.
For $1\leq r\leq n$, $[a,b]\subseteq[1,k]$, $a,b\in\mathbb{N}$ and $\alpha$, denote

$$
\eta_{r,[a,b]}^{\alpha}:=\left\{
\begin{array}{ll}[\frac{\alpha_{br}+\alpha_{b-1,r}}{m_{r}}]\cdots[\frac{\alpha_{a+1,r}+\alpha_{ar}}{m_{r}}]&\;\;\;a-b\text{ odd}\\
[\frac{\alpha_{br}+\alpha_{b-1,r}}{m_{r}}]\cdots[\frac{\alpha_{a+2,r}+\alpha_{a+1,r}}{m_{r}}]\alpha_{ar}&\;\;\;a-b\text{ even}
\end{array}\right.
$$
The following $\omega\in \text{Hom}_{\mathbb{Z}G}(B_k,\mathbb{k}^*)$
\begin{eqnarray}
&&\omega([g_1^{\alpha_{11}}\cdots g_n^{\alpha_{1n}},\dots,g_1^{\alpha_{k1}}\cdots g_n^{\alpha_{kn}}])\\\notag
&=&\prod_{l=1}^{k}\prod_{\begin{array}{ccc}1\leq r_{1}<\cdots<r_{l}\leq n\\\lambda_1+\cdots+\lambda_l=k,\lambda_1\text{ odd}\\\lambda_i\ge1\text{ for }1\le i\le l\end{array}}\zeta_{m_{r_1}}^{(-1)^{\sum_{1\leq i<j\leq l}\lambda_{i}\lambda_{j}}\eta_{r_{1},[a_{1},b_{1}]}^{\alpha}\cdots\eta_{r_{l},[a_{l},b_{l}]}^{\alpha}a_{r_1^{\lambda_1}\cdots r_l^{\lambda_l}}}
\end{eqnarray}
where $a_{l}=1,b_{l}=\lambda_{l},\dots,a_{1}=\lambda_{2}+\cdots+\lambda_{l}+1,b_{1}=\lambda_{1}+\cdots+\lambda_{l}=k$ and $0\leq a_{r_1^{\lambda_1}\cdots r_l^{\lambda_l}}<m_{r_1}$ for $1\leq r_1<\cdots<r_l\leq n$
makes a complete set of representatives of $k$-cocycles of the complex $(B_{\bullet}^*,\partial_{\bullet}^*)$.
Moreover,
$$\text{H}^k(G,\mathbb{k}^*)=\prod_{r=1}^n\mathbb{Z}_{m_r}^{\sum_{j=1}^k(-1)^{k+j}\binom{n-r+j-1}{j-1}}.$$

explicit formulas for them(i.e. people already computed in the literature? such as my answer below)? n-cocycles need to solve cocycle conditions, so are there known results in any Ref? $\endgroup$9more comments