but not necessarily injective map $\Phi : {\tilde}M(G,A) \to {\mathrm{Cent}}(G,A)$ induced from sending a sequence as in to the isomorphism class of $H$.

Viewing $A$ as a trivial $G$-module, we have a bijection between ${\tilde}M(G,A)$ and $H^2(G,A)$, with the class $0 \in H^2(G,A)$ corresponding to all split sequences in $M(G,A)$. We can use this to help determine minimal lifts of irreducible subgroups of $\operatorname{PGL}_n({\mathbb C})$. We recall $H_1(G,{\mathbb Z})$ is the abelianization of $G$, and $H_2(G,{\mathbb Z})$ is the Schur multiplier of $G$.

\[prop:31\] Let $G$ be an irreducible subgroup of $\operatorname{PGL}_n({\mathbb C})$. Then any minimal lift of $G$ to $\operatorname{GL}_n({\mathbb C})$ is a central extension of $G$ by $C_m$ for some divisor $m$ of the exponent of $H_1(G,{\mathbb Z}) \times H_2(G,{\mathbb Z})$.

Any lift of $G$ to an irreducible subgroup $H \subset \operatorname{GL}_n({\mathbb C})$ corresponds to an element of ${\mathrm{Cent}}(G,A)$ where $A=C_m$ for some $m$, and thus corresponds to at least one element of $H^2(G,A)$. The universal coefficients theorem gives us the exact sequence $$\label{eq:uct}
 0 \to \mathrm{Ext}(H_1(G,{\mathbb Z}), A) \to H^2(G, A) \to \mathrm{Hom}(H_2(G,{\mathbb Z}), A) \to 0.$$ Let $m'$ be the gcd of $m$ with the exponent of $H_1(G,{\mathbb Z}) \times H_2(G,{\mathbb Z})$. Recall that $\mathrm{Ext}(\bigoplus {\mathbb 