  Respecting the group theoretical approach, it is discussed that the linear
conformal gravity can be written in terms of a mixed symmetry tensor field of
rank-3 \cite{binegar}. Following this path, related field equation was obtained
in de Sitter space \cite{fatemi}. It was proved that this kind of tensor field
in de Sitter background associates with two unitary irreducible representations
(UIR) of the de Sitter group. The important fact is that one of the them has a
flat limit, namely, in zero curvature coincides to the UIR of Poincar\'{e}
group which has been studied in Ref \cite{rahbar}.
  However, the second one which is named as auxiliary field, becomes
significant in the study of conformal gravity in de Sitter background. In this
paper we will study the second representation in details by finding its
solutions and also the conformally invariant two-point function.
