Order-twelve dicyclic character table ($a^6=1,b^2=a^3,ba{b}^{-1}=a^{-1}$)

This dicyclic group has classes $1$, $a^3$, $\{a,a^5\}$, $\{a^2,a^4\}$ and the two parity classes of $b{a}^j$, of sizes $1,1,2,2,3,3$. Its quotient by $⟨a^2⟩$ is cyclic of order four, supplying four linear characters with $a\mapsto (-1{)}^j$, $b\mapsto i^j$. Its two remaining irreducible representations have $a\mapsto \mathrm{diag}({\zeta }^k,{\zeta }^{-k})$ and $b\mapsto \left(\begin{array}{cc}0& (-1{)}^k\\ 1& 0\end{array}\right)$ for $k=1,2$, where $\zeta =e^{i\pi /3}$. The traces are $2\cos (km\pi /3)$ on $a^m$ and zero on $b{a}^m$. Distinct diagonal eigenlines are exchanged by $b$, proving irreducibility; the degree-square sum $4+4+4=12$ proves completeness.