In 3-dimensions the $\phi^6$ interaction is renormalizable and the $\beta$-function can be found in many reviews in the $O(n)$ symmetric case, $V(\phi) \sim (\phi_a \phi_a)^3$ where $a=1,\ldots,n$.
What I couldn't find, but I can't believe it's not known, is the $\beta$-function in the general (non-symmetric) case:
$$V(\phi) = \frac{1}{6!} \lambda_{abcdef} \phi_a \phi_b \phi_c \phi_d \phi_e \phi_f$$
where the coupling, $\lambda_{abcdef}$, is totally symmetric under the interchange of indices $abcdef$ and so is the corresponding $\beta$-function, $\beta_{abcdef}$. I imagine the result is known to at least a couple of loops, probably at least 4 or 5.
