  We perform a comprehensive perturbative study of the operator spectrum in
multi-scalar theories with hypercubic global symmetry. This includes working
out symmetry representations and their corresponding tensor structures. These
structures are then used to compute the anomalous dimensions of scalar
operators with up to four fields and arbitrary representations to six-loop
order. Moreover, we determine one-loop anomalous dimensions for a large number
of low-lying operators in the spectrum which include more powers of the
fundamental field and/or insertions of derivatives. As an aside we show how
projectors used in the conformal bootstrap can be conveniently reused in
computations of anomalous dimensions. The results of our study are of use to
the conformal bootstrap. They also illuminate features of conformal
perturbation theory and the large $n$ expansion. Our results may be of interest
for various crossover phenomena in statistical field theory. In total, we
compute the scaling dimension of more than 300 operators, of which 16 are
computed to six-loops. Our analysis is exhaustive with respect to group theory
up to rank 4 for any number of flavours $n$, and also exhaustive with respect
to which representations exist for $n \leq 4$.
