There is folklore theorem concerning dimensions of complex projective representations of nilpotent groups that I want a reference for. I have searched Karpilovsky monographs but no success. The theorem is a combination of two known claims.
Claim1 (known): Let $p$ be a prime number, let $P$ be a $p$-group and let $\alpha \in Z^2(P,\mathbb{C}^*)$ be a $2$-cocycle. Then there exists an irreducible $\alpha$-projective representation of $P$ such that it's dimension divides the dimension of any other irreducible $\alpha$-projective representation of $P$.
Claim2 (known): Let $G$ be a group such that $G=H_1\times H_2$ and assume that $\alpha_1 \in Z^2(H_1,\mathbb{C}^*)$, $\alpha_2 \in Z^2(H_2,\mathbb{C}^*)$ and assume $\alpha=\alpha_1\cdot \alpha_2$, that is $\alpha \in Z^2(G,\mathbb{C}^*)$ is a $2$ cocycle such that $\alpha (h_1h_2,k_1k_2)=\alpha_1(h_1,k_1)\cdot \alpha_2(h_2,k_2)$ for any $h_1,k_1\in H_1$ and $h_2,k_2\in H_2$. Then the dimensions of the $\alpha$-projective representations are just multiplications of the dimensions of the $\alpha _1$-projective representation of $H_1$ with the dimensions of the $\alpha _2$-projective representations of $H_2$.
I am looking for a reference for both claims or the consequence of both claims for nilpotent groups.
Any help will be appreciated.