  We have computed the Clebsch-Gordan coefficients for the product (000001)
$\otimes$ (000001), where (000001) is the adjoint 78-dimensional representation
of $E_6$. The results are presented for the dominant weights of the irreducible
representations in this product. As a simple application we express the singlet
operator in $ 27 \otimes 78 \otimes \bar{27}$ in terms of multiplets of the
Standard Model gauge group.
