= Solution
Every orientation-preserving isometry of the <Poincaré half-plane model> has the form
$$
\boxed{z\longmapsto\frac{az+b}{cz+d}},
\qquad a,b,c,d\in\mathbb R,\quad ad-bc=1,
$$
with matrices identified up to multiplication by $-I$. Thus $G\cong PSL(2,\mathbb R)$.
For $z=x+iy\in\mathbb H$, the affine map $(z-x)/y$ sends $z$ to $i$, proving transitivity on points. Real translations and positive dilations act transitively on finite boundary points, while inversion $z\mapsto-1/z$ exchanges $0$ and $\infty$; hence the action on $\mathbb R\cup\{\infty\}$ is transitive. A hyperbolic line is determined by its unordered pair of boundary endpoints, and a real Möbius transformation can send any such pair to $\{0,\infty\}$. Therefore $G$ is also transitive on hyperbolic lines.
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