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