= Order-twelve dicyclic character table
{title2=$a^6=1,\ b^2=a^3,\ bab^{-1}=a^{-1}$}
This <dicyclic group> has classes $1$, $a^3$, $\{a,a^5\}$, $\{a^2,a^4\}$ and the two parity classes of $ba^j$, of sizes $1,1,2,2,3,3$. Its quotient by $\langle a^2\rangle$ is cyclic of order four, supplying four linear characters with $a\mapsto(-1)^j$, $b\mapsto i^j$. Its two remaining irreducible representations have $a\mapsto\operatorname{diag}(\zeta^k,\zeta^{-k})$ and $b\mapsto\left(\begin{smallmatrix}0&(-1)^k\\1&0\end{smallmatrix}\right)$ for $k=1,2$, where $\zeta=e^{i\pi/3}$. The traces are $2\cos(km\pi/3)$ on $a^m$ and zero on $ba^m$. Distinct diagonal eigenlines are exchanged by $b$, proving irreducibility; the degree-square sum $4+4+4=12$ proves completeness.
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