Using the complete Galaxy and Mass Assembly I (GAMA-I) survey covering similar to 142 deg(2) to r(AB) = 19.4, of which similar to 47 deg(2) is to r(AB) = 19.8, we create the GAMA-I galaxy group catalogue (G(3)Cv1), generated using a friends-of-friends (FoF) based grouping algorithm. Our algorithm has been tested extensively on one family of mock GAMA lightcones, constructed from Lambda cold dark matter N-body simulations populated with semi-analytic galaxies. Recovered group properties are robust to the effects of interlopers and are median unbiased in the most important respects. G(3)Cv1 contains 14 388 galaxy groups (with multiplicity >= 2), including 44 186 galaxies out of a possible 110 192 galaxies, implying similar to 40 per cent of all galaxies are assigned to a group. The similarities of the mock group catalogues and G(3)Cv1 are multiple: global characteristics are in general well recovered. However, we do find a noticeable deficit in the number of high multiplicity groups in GAMA compared to the mocks. Additionally, despite exceptionally good local spatial completeness, G(3)Cv1 contains significantly fewer compact groups with five or more members, this effect becoming most evident for high multiplicity systems. These two differences are most likely due to limitations in the physics included of the current GAMA lightcone mock. Further studies using a variety of galaxy formation models are required to confirm their exact origin. The G(3)Cv1 catalogue will be made publicly available as and when the relevant GAMA redshifts are made available at http://www.gama-survey.org.