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The Weyl dimension formula is increasing in each nonnegative Dynkin label. If $a\ge 3$ or $b\ge 3$, the dimension is at least $d(3,0)=10$. Among $0\le a,b\le 2$, the cases with both labels nonzero, other than $(1,1)$, have dimension at least $d(2,1)=15$. Thus the following is the complete list of irreducible representations of dimension strictly less than $10$:Here $6={\mathrm{Sym}}^{2}3$ is the two-index symmetric representation and $8$ is the Adjoint representation. Interchanging $a$ and $b$ gives the conjugate representation, explaining equal dimensions, Casimir eigenvalues and Dynkin indices in each conjugate pair. In the alternative convention of solution, divide the $C$ and $c$ columns by two.

The two-index symmetric representation of SU(3) is $6$, with Dynkin index $5$. Thus $n$ copies have $c(r_S)=5n$, and the supplied beta function isThe strict sign condition for asymptotic freedom is $n<132/5=26.4$, givingIndeed $132-5(26)=2>0$, while $132-5(27)=-3<0$.

For the fundamental representation and the two-index symmetric representation, the Dynkin indices are $1$ and $5$. Therefore the specified discrete gauge anomaly condition for ${\mathbb{Z}}_4$ isAll ordered solutions areOnly $(1,3)$ and $(3,1)$ give a faithful ${\mathbb{Z}}_4$ action on this pair of Weyl fermions; faithfulness was not imposed. As in solution, cancellation here concerns the stipulated discrete gauge anomaly, not cancellation of the pure cubic SU(3) gauge anomaly. The integer normalization matters: one must not replace $c$ by a half-integer trace index inside an unchanged modular congruence.

The composite ${\chi }^{ij}$ is in the two-index symmetric representation of $SU(p)$ and hasFor that representation,Its anomalies are consequentlyin the same order as part iii. Every anomaly matches, so confinement to the proposed massless composite is consistent with 't Hooft anomaly matching.