A systematic study is made of the irreducible representations of the unitary unimodular group in three dimensions, an algebraic technique being used that is based on the well-known theory of the rotation group in three dimensions. Explicit representation matrices are found. The representations are characterized by the ranges of the `hypercharge' and `isobaric spin' that occur in them. The dimension formula is derived and the invariants are determined. Two identities are found relating the generators of the group.