  We investigate the stability of synchronized states in delay-coupled networks
where synchronization takes place in groups of different local dynamics or in
cluster states in networks with identical local dynamics. Using a master
stability approach, we find that the master stability function shows a discrete
rotational symmetry depending on the number of groups. The coupling matrices
that permit solutions on group or cluster synchronization manifolds show a very
similar symmetry in their eigenvalue spectrum, which helps to simplify the
evaluation of the master stability function. Our theory allows for the
characterization of stability of different patterns of synchronized dynamics in
networks with multiple delay times, multiple coupling functions, but also with
multiple kinds of local dynamics in the networks' nodes. We illustrate our
results by calculating stability in the example of delay-coupled semiconductor
lasers and in a model for neuronal spiking dynamics.
