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Let $G$ be a finite abelian group, and $\alpha \in H^2 (G,\mathbb{C}^*)$ a 2-cohomology class.

It is known (in Karpilovsky's multi-volume tome or elsewhere) that a finite abelian group admits a faithful irreducible projective representation if and only if it is of symmetric type i.e. $G=H\times H$ for some group $H$. Furthermore, all faithful irreducible $\alpha$-representations of $G$ are projectively equivalent and have degree $\sqrt{\left|G \right|}$.

My question is how to count the number of all 2-cohomology classes that give rise to faithful irreducible projective representations of $G$ and construct explicit matrix representations of all of the representations.

If there is no good answer in general cases, I would be content with the groups $\mathbb{Z}_4 \times \mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. I guess (I'm not familiar with computational group theory tools) that a straightforward way is to search for covering groups of those groups and then check their irreducible representations.

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