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The renormalization group functions for six dimensional scalar $\phi^3$theory with an $F_4$ symmetry are provided at four loops in the modifiedminimal subtraction (MSbar) scheme. Aside from the anomalous dimension of$\phi$ and the $\beta$-function this includes the mass operator and a$\phi^2$-type operator. The anomalous dimension of the latter is computedexplicitly at four loops for the $\mathbf{26}$ and $\mathbf{324}$representations of $F_4$. The $\epsilon$ expansion of all the related criticalexponents are determined to $O(\epsilon^4)$. For instance the value for$\Delta_\phi$ agrees with recent conformal bootstrap estimates in $5$ and$5.95$ dimensions. The renormalization group functions are also provided atfour loops for the group $E_6$.

$F_4$ symmetric $\phi^3$ theory at four loops