We consider a setting where a large number of agents are all interested in attending some public resource of limited capacity. Attendance is thus allotted by lottery. If agents arrive individually, then randomly choosing the agents – one by one - is a natural, fair and efficient solution. We consider the case where agents are organized in groups (e.g. families, friends), the members of each of which must all be admitted together. We study the question of how best to design such lotteries. We first establish the desired properties of such lotteries, in terms of fairness and efficiency, and define the appropriate notions of strategy proofness (providing that agents cannot gain by misrepresenting the true groups, e.g. joining or splitting groups). We establish inter-relationships between the different properties, proving properties that cannot be fulfilled simultaneously (e.g. leximin optimality and strong group stratagy proofness). Our main contribution is a polynomial mechanism for the problem, which guarantees many of the desired properties, including: leximin optimality, Pareto-optimality, anonymity, group strategy proofness, and adjunctive strategy proofness (which provides that no benefit can be obtained by registering additional - uninterested or bogus - individuals). The mechanism approximates the utilitarian optimum to within a factor of 2, which, we prove, is optimal for any mechanism that guarantees any one of the following properties: egalitarian welfare optimality, leximin optimality, envyfreeness, and adjunctive strategy proofness.