Projective representations of a finite abelian group

Question

Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups

$$ G\cong C_{n_1,...,n_k}:=\mathbb{Z}_{n_1}\times ...\times \mathbb{Z}_{n_k} $$

Hence I was wandering if there is any known result for

$$ H^2(C_{n_1,...,n_k},U(1)) $$

I know that $$ H^2(\mathbb{Z}_n,U(1))=0 \ , \ \ \ \ \ \ H^2(\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2},U(1))\cong \mathbb{Z}_{\text{gcd}(n_1,n_2)} $$ but I don't know about are a general result or even a result for $k=3$.

Answer 1

The answer is $\ \displaystyle\bigoplus_{i<j}\ \mathbb{Z}/\!\gcd(n_i,n_j)$. The reason is that for $G$ finite, $H^2(G,U(1))$ is the dual abelian group of $H_2(G,\mathbb{Z})$. Now use the Künneth formula.