We present the Complete Local-Volume Groups Sample (CLoGS), a statistically complete optically-selected sample of 53 groups within 80 Mpc. Our goal is to combine X-ray, radio and optical data to investigate the relationship between member galaxies, their active nuclei, and the hot intra-group medium (IGM). We describe sample selection, define a 26-group high-richness subsample of groups containing at least 4 optically bright (log L_B>=10.2 LBsol) galaxies, and report the results of XMM-Newton and Chandra observations of these systems. We find that 14 of the 26 groups are X-ray bright, possessing a group-scale IGM extending at least 65kpc and with luminosity >10^41 erg/s, while a further 3 groups host smaller galaxy-scale gas halos. The X-ray bright groups have masses in the range M_500=0.5-5x10^13 Msol, based on system temperatures of 0.4-1.4 keV, and X-ray luminosities in the range 2-200x10^41 erg/s. We find that ~53-65% of the X-ray bright groups have cool cores, a somewhat lower fraction than found by previous archival surveys. Approximately 30% of the X-ray bright groups show evidence of recent dynamical interactions (mergers or sloshing), and ~35% of their dominant early-type galaxies host AGN with radio jets. We find no groups with unusually high central entropies, as predicted by some simulations, and confirm that CLoGS is in principle capable of detecting such systems. We identify three previously unrecognized groups, and find that they are either faint (L_X,R500<10^42 erg/s) with no concentrated cool core, or highly disturbed. This leads us to suggest that ~20% of X-ray bright groups in the local universe may still be unidentified.
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