What is the spacetime symmetry group of special relativity including translations?

The spacetime symmetry group of special relativity, including translations, is the Poincaré group.

Special relativity describes flat Minkowski spacetime. Its full set of distance-preserving transformations includes translations through space and time, spatial rotations, and Lorentz boosts between uniformly moving observers. Together, these transformations form the Poincaré group.

A common mix-up is to answer “Lorentz group.” The Lorentz group includes rotations and boosts but leaves the coordinate origin fixed; adding the four spacetime translations produces the larger Poincaré group. In four dimensions, the group has 10 continuous parameters: four translations, three rotations, and three boosts.

The group is named after Henri Poincaré, although Hermann Minkowski formulated the relevant spacetime geometry in 1908. In quantum field theory, Poincaré symmetry also organizes particle states by quantities such as mass and spin. In general relativity, it applies globally only in flat spacetime, while curved spacetimes generally retain it only locally.