A cover is a family of 6-element subsets, called blocks, chosen from a 12-element universe, such that each 3-element subset is contained in at least one block. This paper constructs a cover with 15 blocks, and it shows that any cover has at least 15 blocks; thus the covering number . It also shows that the 68 nonisomorphic covers with 15 blocks fall into just two classes using a very natural classification scheme.
