There are two main results in this paper. First, it is shown that we can develop a theory of classes in close analogy to the usual theory of representations. We can introduce concepts, such as reducible and irreducible classes, sum and product of classes, reduction of a class when going from a group to a subgroup, etc. Second, it is shown that it is possible to associate a ``magic square'' to each group. It is related to the numbers of pairs of commuting elements between classes and it can be used immediately to find the structure of the ``tensor operators'' of the group.