Dynamical system identification aims to recover system parameters from observed trajectories, and in scientific settings those dynamics often obey unknown symmetries. This work studies how to identify an equivariant dynamical system from a single trajectory when the underlying symmetry group is not known in advance. It first characterizes the shorter trajectory lengths needed when the symmetry is known, then proposes a method that learns the symmetry group from data and incorporates it into identification. The analysis uses tools from group representation theory and Cayley graph expansion.