Test generation and execution are often hampered by the large state spaces of the systems involved. In automata (or transition system) based test algorithms, taking advantage of symmetry in the behavior of speci cation and implementation may substantially reduce the amount of tests. We present a framework for describing and exploiting symmetries in black box test derivation methods based on nite state machines (FSMs). An algorithm is presented that, for a given symmetry relation on the traces of an FSM, computes a subautomaton that characterizes the FSM up to symmetry. This machinery is applied to the classical W-method 26, 7] for test derivation. Finally, we focus on symmetries de ned in terms of repeating patterns. 1991 Mathematics Subject Classi cation: 68M15, 68Q05, 68Q68, 94C12