The number of distinct latin squares as a group-theoretical constant

It is shown that the number l_n of all distinct Latin squares of the nth order appears as a structure constant of the algebra defined on the Magic squares of the same order. The algebra is isomorphic to the algebra of double cosets of the symmetric group of degree n² with respect to the intransitive subgroup of all substitutions in the n sets of transitivity, each set being of cardinality n. The representation theory makes it possible then to express l_n in terms of eigenvalues of a certain element of the algebra.