  An important aspect in the evolutionary scenario of cool stars is their
rotation and the rotationally induced magnetic activity and interior mixing.
Stars in open clusters are particularly useful tracers for these aspects
because of their known ages. We aim to characterize the open cluster IC4756 and
measure stellar rotation periods and surface differential rotation for a sample
of its member stars. Thirty-seven cluster stars were observed continuously with
the CoRoT satellite for 78 days in 2010. Follow-up high-resolution spectroscopy
of the CoRoT targets and deep Str\"omgren $uvby\beta$ and H$\alpha$ photometry
of the entire cluster were obtained with our robotic STELLA facility and its
echelle spectrograph and wide-field imager, respectively. We determined
high-precision photometric periods for 27 of the 37 CoRoT targets and found
values between 0.155 and 11.4 days. Twenty of these are rotation periods.
Twelve targets are spectroscopic binaries of which 11 were previously unknown;
orbits are given for six of them. Six targets were found that show evidence of
differential rotation with $\Delta\Omega/\Omega$ in the range 0.04-0.15. Five
stars are non-radially pulsating stars with fundamental periods of below 1d,
two stars are semi-contact binaries, and one target is a micro-flaring star
that also shows rotational modulation. Nine stars in total were not considered
members because of much redder color(s) and deviant radial velocities with
respect to the cluster mean. H$\alpha$ photometry indicates that the cluster
ensemble does not contain magnetically over-active stars. The cluster average
metallicity is -0.08$\pm$0.06 (rms) and its logarithmic lithium abundance for
12 G-dwarf stars is 2.39$\pm$0.17 (rms). [...]
