The physical symmetries of an N-electron gas in a d-dimensional quantum dot are studied. The symplectic group framework is used to discuss the dynamical modes of the system and to classify the N-electron states including spin. We focus on the results for a few-electron (N\ensuremath{\le}12) dot in d=2 dimensions. For N4 electrons the types of configurations appearing are strongly N dependent. The allowed configurations for N4 are also highly restricted compared to those for N\ensuremath{\ge}4. These findings suggest that results of few-electron (N4) cluster calculations may have limited applicability to large-N systems.