We present simulations (P3M + SPH) containing both dark matter and gas, of the formation of a cluster of galaxies. An efficient potential solver, SPH algorithm and cooling implementation make for an extremely fast scheme. The numerical two-body relaxation time is shown to constrain severely the resolution which can be attained. The cluster in our simulations forms by the merger of three large groups at which time relaxation creates an isothermal mass distribution with no central core. Subsequent infall of small groups does not disturb this structure but gives an outer radius which grows uniformly with time. The density and temperature of the gas fall in the outer regions of the cluster but the combination T/n2/3, which is related to the entropy, or shock heating, rises linearly with radius before dropping steeply at the edge of the cluster. The morphology of the cluster is not determined by the initial shape of the overdensity which gives rise to it, but is strongly affected by the large-scale tidal field. The collapse is well fitted by a simple top-hat model up to the point of maximum expansion but is then slowed either by the presence of substructure or because of the non-spherical shape. Press-Schechter theory underestimates the number density of massive clusters. Pre-virialization does not appear to diminish the degree of collapse. Fitting an isothermal beta-model to the density profiles of the cluster and other large groups gives beta(fit) almost-equal-to 2/3. There is incomplete thermalization of the cluster gas but this may well be a numerical artefact arising from inadequate resolution of shocks. After correcting for this, the ratio of dark matter to gas energies is beta almost-equal-to 1. We show that the difference between beta and beta(fit) is consistent with a hydrostatic model. However, this does not explain the long-standing beta-discrepancy which, because of the steeper decline of galaxy density with radius, predicts a ratio of galaxy to gas energies of beta(spec) almost-equal-to beta(fit) contrary to observations: possible resolutions of this problem are discussed. When cooling is added to our simulation the intracluster medium becomes multiphase within the cooling radius and a large quantity of cool gas accumulates at the cluster centre. The gas temperature rises steeply from 10(4) K in low-mass haloes to the virial temperature of 10(7) K or more in high-mass ones, in agreement with simple hierarchical models of galaxy formation. Subgroups can survive within the virialized portion of the cluster. They tend to be on radial orbits with a velocity dispersion (internal plus external) equal to the cluster mean. Mass estimators based on the virial theorem underestimate the total cluster mass and have large variance. A more stable statistic is the projected mass estimator of Heisler, Tremaine & Bahcall. The relationship between the X-ray luminosity and the mass function of clusters remains obscure but where the surface brightness profile can be measured the mass profile of a cluster is well constrained. Finally, we discuss several ways in which the simulations presented in this paper can be extended and improved.
