I know that the group of isometries on $\mathbb H^n(R)$ is $O^+(n,1)$ (the orthochronous lorentz group) which i have proved using the Hyperboloid model.
I have also been able to show for n=2 case considering half plane model the isometies are composition of Mobius transformation and reflection (with respect to the imaginary axis). Also I have shown that the isometry group is isomorphic to $PSL(2,\mathbb R)\rtimes_\alpha\frac{\mathbb Z}{2\mathbb Z}$, where $$\alpha :\frac{\mathbb Z}{2\mathbb Z} \to \text{Aut}(PSL(2,\mathbb R))$$ $$\alpha(1)=\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\mapsto \begin{pmatrix} a & -b \\ -c & d \\ \end{pmatrix}$$
But I am unable to link this two informations for n=2. Is there a way to see a similar thing in higher dimensions? if possible suggest some reference regarding representations of $O^+(n,1)$, as I don't understand much about them.