  We look for chiral primaries in the general Leigh-Strassler deformed N=4
super Yang-Mills theory by systematically computing the planar one-loop
anomalous dimension for single trace operators up to dimension six. The
operators are organised into representations of the trihedral group,
\Delta(27), which is a symmetry of the Lagrangian. We find an interesting
relationship between the U(1)_R-charge of chiral primaries and the
representation of \Delta(27) to which the operator belongs. Up to scaling
dimension \Delta_0=6 (and conjecturally to all dimensions) the following holds:
The planar one-loop anomalous dimension vanishes only for operators that are in
the singlet or three dimensional representations of \Delta(27). For other
operators, the vanishing of the one-loop anomalous dimension occurs only in a
sub-locus in the space of couplings.
