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Define $G_C := G \times_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1)$ by endowing the Cartesian product $G \times \mathrm{U}(1)$ with the multiplication $$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$ One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \times_C\mathrm{U}(1)$ is in the center of $G_C$, and $$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$ is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation $$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$ Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.

Define $G_C := G \times_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1))$ by endowing the Cartesian product $G \times \mathrm{U}(1)$ with the multiplication $$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$ One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \times_C\mathrm{U}(1)$ is in the center of $G_C$, and $$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$ is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation $$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$ Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.

Define the semi-direct product $G_C := G \ltimes_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1)$ by endowing the Cartesion product $G \times \mathrm{U}(1)$ with the multiplication $$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$ One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \ltimes_C\mathrm{U}(1)$ is in the center of $G_C$, and $$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$ is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation $$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$ Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.

Define $G_C := G \times_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1)$ by endowing the Cartesian product $G \times \mathrm{U}(1)$ with the multiplication $$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$ One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \times_C\mathrm{U}(1)$ is in the center of $G_C$, and $$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$ is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation $$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$ Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.

Define the central extension $G_C := G \times_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1)$ by endowing the Cartesian product $G \times \mathrm{U}(1)$ with the multiplication $$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$ One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \times_C\mathrm{U}(1)$ is in the center of $G_C$, and $$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$ is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation $$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$ Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.

Define the semi-direct product $G_C := G \ltimes_C \mathrm{U}(1)$ for any representative $C$ of an element in $H^2(G,\mathrm{U}(1)$ by endowing the Cartesion product $G \times \mathrm{U}(1)$ with the multiplication $$ (g,\alpha)\cdot(h,\beta) := (gh,\alpha\beta C(g,h))$$ One may check that it is a central extension, i.e. the image of $\mathrm{U}(1)\to G \ltimes_C\mathrm{U}(1)$ is in the center of $G_C$, and $$ 1 \to \mathrm{U}(1) \to G_C \to G \to 1$$ is exact. For any projective representation $\sigma$, fix $\Sigma,C$ and define the linear representation $$ \sigma_C : G_C \to \mathrm{U}(\mathcal{H}), (g,\alpha) \mapsto \alpha\Sigma(g)$$ Conversely, every unitary representation $\rho$ of some $G_C$ gives a pair $\Sigma,C$ by $\Sigma(g) = \alpha^{-1}\rho(g,\alpha)$.