Isometries and inner product in Hyperbolic SPACE (H3)

Well, the title says what I need to know/understand, when I studied upper half plane I remember that isometries are Mobius transformations (If I am not wrong), now I have no clue about it. Thanks for your help!

With the upper-half space model of hyperbolic $3$-space, it is easy to see that the orientation-preserving isometry group of $\mathbb{H}^3$ is $PSL(2,\mathbb{C})$, as was first observed by Poincare. The Moebius transformations $$z\mapsto \frac{az+b}{cz+d}$$ from $\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ extend to an isometry of $\mathbb{H}^3$. Furthermore $PSL(2,\mathbb{\mathbb{C}})$ is of index $2$ in the full isometry group $Isom(\mathbb{H}^3)\cong PSL(2,\mathbb{C})\rtimes \mathbb{Z}/2\mathbb{Z}$, where the nontrivial element of $\mathbb{Z}/2\mathbb{Z}$ acts by complex conjugation on $PSL(2,\mathbb{C})$. A very good reference is the book Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory, by Juergen Elstrodt, Fritz Grunewald and Jens Mennicke.
• I saw that $PSL(2,\mathbb{C})$ acts on the boundary of the upper half space by $$\left(\begin{array}{cc}a&b\\c&d\end{array}\right).z=\frac{az+b}{cz+d}$$ and it can be extended to interior of $\mathbb{H}^3$. How can it be extended?
• I Read about the Poincare extension, for $Isom^+(\mathbb{H}^3)$ But I really don´t know what you mean when you wrote $Isom(\mathbb{H}^3)\cong PSL(2,\mathbb{C})\rtimes \mathbb{Z}/2\mathbb{Z}$. What is $\rtimes$?