Explain why $\newcommand{\Z}{\mathbb{Z}} U\_{44} \cong (\Z\_{10} \oplus \Z\_2) $ 

I know that $\Z\_{20} \cong (\Z\_{10} \oplus \Z\_2)$, so if I can show $U\_{44} \cong \Z\_{20}$, then I can conclude that $U\_{44} \cong (\Z\_{10} \oplus \Z\_2) $ since isomorphism is transitive.

In order to show $U\_{44} \cong \Z\_{20}$, I need to show that the function $ \varphi: U\_{44} \to \Z\_{20}$ is bijective and structure preserved. I know that $U\_{44} \text{ and } \Z\_{20}$ have order $20$, so the function can be bijective, but it's not enough to show $\varphi: U\_{44} \to \Z\_{20} $ is bijective.

I wonder if anyone can show me how to show $ \varphi: U\_{44} \to \Z\_{20} $ is bijective and structure preserving.