A 10 dimensional Lie group, the relativistic counterpart of which is the 10 dimensional Poincare group. Both include the isometry group of Euclidean space. Moreover, both have important linear representations including actions on tangents of trajectories (i.e. velocities), which - in case of the Galilean group - shift all but one (i.e. the spatial) components like a translation.
Wikipedia, Galilean group
Wikipedia, Euclidean group
David R. Hilbert, Nick Huggett, Groups in Mind, Philosophy of Science 73 (5):765-777 (2006) (doi:10.1086/518635, pdf)
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