Rayleigh–Bénard convection in a finite rotating cylinder of moderate aspect ratio (radius four times the depth) is investigated numerically for a fluid of Prandtl number equal to 7 (corresponding essentially to water). We consider the effects of rotation from both the Coriolis force and the centrifugal force and find that the centrifugal force plays a significant dynamic role. In this initial study, we restrict the computations to the axisymmetric subspace in which the convection patterns near onset consist of steady concentric circular cells, the so-called target patterns, which have been studied and observed experimentally under different conditions by a number of investigators. As the convection is driven far enough beyond onset, the steady cellular patterns give way to time-periodic states in which the target patterns travel radially inward. We have identified two such travelling modes, primarily distinguished by one having alternating warm and cold plumes forming at the cylinder sidewall and then propagating radially inward to quench alternately cold and warm plumes on the axis. The other mode always has a cold plume descending on the sidewall and the adjacent warm plume periodically splits into two, with the innermost of the split pair travelling radially inward. The first of these modes is found when the centrifugal force is weak and the second for stronger centrifugal force. The large-scale meridional circulation driven by the centrifugal buoyancy is seen to favour having a cold plume descending on the sidewall, accounting for the switch to the second travelling mode.