  Requiring that supersymmetric SU(5) and SU(6) grand unifications in the
heterotic string theory must have 3 chiral families, adjoint (or higher
representation) Higgs fields in the grand unifiedgauge group, and a non-abelian
hidden sector, we construct such string models within the framework of
conformal field theory and asymmetric orbifolds. Within this framework, we
construct all such string models via Z_6 asymmetric orbifolds that include a
Z_3 outer-automorphism, the latter yielding a level-3 current algebra for the
grand unification gauge group SU(5) or SU(6). We then classify all such Z_6
asymmetric orbifolds that result in models with a non-abelian hidden sector.
All models classified in this paper have only one adjoint (but no otherhigher
representation) Higgs field in the grand unified gauge group. This Higgs field
is neutral under all other gauge symmetries. The list of hidden sectors for
3-family SU(6) string models are SU(2), SU(3) and $SU(2) \otimes SU(2)$. In
addition to these, 3-family SU(5) string models can also have an SU(4) hidden
sector. Some of the models have an anomalous U(1).
