the automorphism group of the disk
Collect together every conformal automorphism of the disk — every holomorphic, one-to-one, onto self-map — and you get a set that is closed under composition (do one, then another, and you are still in the set) and under taking inverses. That makes it a group: the automorphism group of the disk, written Aut(D). It is the complete catalogue of the disk's holomorphic symmetries, the full list of rigid motions of the disk's hyperbolic geometry.
Every element has the standard form f(z) = e^(i theta) (z - a)/(1 - a-bar z), with |a| < 1 and theta real, so the group is parametrized by three real numbers: where the center goes (the point a, two coordinates) and a final rotation angle theta. Composing two such maps gives another of the same form — you can check this is a genuine group operation, with identity the map z -> z and inverses given by another automorphism. Structurally, Aut(D) is isomorphic to the matrix group PSU(1,1) (2-by-2 complex matrices preserving a certain indefinite form, modulo scalars), and via the Cayley transform it is isomorphic to PSL(2,R), the automorphism group of the upper half-plane. So 'disk symmetries' and 'upper-half-plane symmetries' are the same abstract group wearing two costumes.
This group is the reason the disk deserves to be called a homogeneous geometry: it acts transitively (move any point to any other) and the point stabilizers are the rotations, so the disk is the quotient (the homogeneous space) of the group by a rotation subgroup. It is precisely the orientation-preserving isometry group of the Poincare metric. It is also the symmetry group underlying hyperbolic tilings, modular forms (through its discrete subgroups like the modular group), and the spectral analysis of the disk. An honest scope note: Aut(D) here means the HOLOMORPHIC automorphisms. Including anticonformal reflections gives a group twice as big, but those reverse orientation and live outside this holomorphic story.
Composition stays inside the group: rotate by theta (z -> e^(i theta) z), then recenter via a Blaschke factor B_a. The result is again of the form e^(i phi)(z - b)/(1 - b-bar z) for some new b and phi. In particular the subgroup of pure rotations {z -> e^(i theta) z} is exactly the stabilizer of the center 0, since those are the only automorphisms fixing 0 (that is the Schwarz lemma talking).
Rotations are the stabilizer of 0; together with the Blaschke recenterings they generate all of Aut(D).
Aut(D) is three-dimensional as a real group, NOT the full six-real-dimensional Mobius group: only the Mobius maps preserving the disk qualify. Beware too that it is non-abelian — composing a rotation and a Blaschke recentering in the two orders gives different results.