= Solution
The <Dynkin index> is additive over a <direct sum of group representations>, so $n$ copies of the <fundamental representation> have $c(r_S)=n$. With $c(\mathrm{ad})=6$, the supplied <beta function (physics)> becomes
$$
\beta(g)=-\frac{g^3}{192\pi^2}(132-n).
$$
For positive <gauge coupling> $g$, <asymptotic freedom> at this order requires $\beta(g)<0$, hence $n<132$. For nonnegative integer $n$,
$$
\boxed{n_{\max}=131.}
$$
At $n=132$ the one-loop coefficient vanishes, so this formula alone does not establish <asymptotic freedom>. This count follows the scalar multiplicity convention of the printed <beta function (physics)>; changing how a real or complex <scalar field> is counted requires changing its coefficient too.
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