Let $G$ be an abelian group and let $\varphi:Z_2\rightarrow Aut(G)$ be the homomorphism where $\varphi(\overline{1})$ is the inversion automorphism.
Define $Dih(G)=G\rtimes_{\varphi} Z_2$.
Now set $G=Z_n$.
Then, does $(\overline{1}_n,0)$ represent the rotation in $D_n$ and $(0,\overline{1}_2)$ represent the reflection in $D_n$?
Moreover, the usual description of $D_n$ in abstract algebra texts are really informal. Even though $D_n$ can be defined using $O(n)$, I think the definition by semidirect product is really easy and natural, however I'm not sure whether this is the standard formal definition. What is the standard formal definition of $D_n$?