  We calculate the vacuum stability conditions and renormalisation group
equations for the extensions of standard model with a higher colour multiplet
scalar up to the representation $\mathbf{15'}$ that leaves the strong
interaction asymptotically free. In order to find the vacuum stability
conditions, we calculate the orbit spaces for the self-couplings of the higher
multiplets, which for the representations $\mathbf{15}$ and $\mathbf{15'}$ of
$SU(3)_c$ are highly complicated. However, if the scalar potential is linear in
orbit space variables, it is sufficient to know the convex hull of the orbit
space. In contrast to the self-couplings of other multiplets, we find that the
scalar quartic couplings of the representations $\mathbf{3}$ and $\mathbf{8}$
walk rather than run, remaining nearly constant and perturbative over a vast
energy range. We describe the conditions for walking couplings using a
schematic model. With these technical results at hand we revise earlier results
of generation of new scales with large $SU(3)_c$ scalar multiplets. Our results
are easily extendable to models of new physics with additional $SU(3)$ or
SU($N$) gauge symmetries.
