  In an effort to develop tools for grand unified model building for the Lie
group $E_6$, in this paper we present the computation of the Clebsch-Gordan
coefficients for the product (100000) $\otimes$ (000010), where (100000) is the
fundamental 27-dimensional representation of $E_6$ and (000010) is its charged
conjugate. The results are presented in terms of the dominant weight states of
the irreducible representations in this product. These results are necessary
for the group analysis of $E_6$ operators involving also higher
representations, which is the next step in this project. In this paper we apply
the results to the construction of the operator ${\bf 27}^3$.
