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We consider $\phi^3$ theory in $6-2\epsilon$ with $F_4$ global symmetry. Thebeta function is calculated up to 3 loops, and a stable unitary IR fixed pointis observed. The anomalous dimensions of operators quadratic or cubic in $\phi$are also computed. We then employ conformal bootstrap technique to study thefixed point predicted from the perturbative approach. For each putative scalingdimension of $\phi$ ($\Delta_{\phi})$, we obtain the corresponding upper boundon the scaling dimension of the second lowest scalar primary in the ${\mathbf26}$ representation $(\Delta^{\rm 2nd}_{{\mathbf 26}})$ which appears in theOPE of $\phi\times\phi$. In $D=5.95$, we observe a sharp peak on the upperbound curve located at $\Delta_{\phi}$ equal to the value predicted by the3-loop computation. In $D=5$, we observe a weak kink on the upper bound curveat $(\Delta_{\phi},\Delta^{\rm 2nd}_{{\mathbf 26}})$=$(1.6,4)$.

$\phi^3$ theory with $F_4$ flavor symmetry in $6-2\epsilon$ dimensions: 3-loop renormalization and conformal bootstrap