{c}
{title2=$G_{6n}$}

For
$$
G_{6n}=\langle a,b:a^{2n}=b^3=1,\ a^{-1}ba=b^{-1}\rangle,
$$
there are $2n$ <linear characters>, given by $a\mapsto\xi^j$, $b\mapsto1$ for a primitive $(2n)$th <root of unity> $\xi$, and $n$ two-dimensional irreducible representations
$$
a\mapsto\begin{pmatrix}0&\xi^k\\\xi^k&0\end{pmatrix},
\qquad
b\mapsto\begin{pmatrix}\omega&0\\0&\omega^2\end{pmatrix},
\qquad 0\leq k<n,
$$
where $\omega^3=1$ and $\omega\ne1$. The two-dimensional representations parametrized by $\varepsilon$ and $-\varepsilon$ are isomorphic. The <sum of squares of irreducible degrees> is $2n+4n=6n$, so this list is complete.


 Irreducible representations of G6n

ID: irreducible-representations-of-g6n