Irreducible representations of G6n ($G_{6n}$)

Forthere are $2n$ linear characters, given by $a\mapsto {\xi }^j$, $b\mapsto 1$ for a primitive $(2n)$th root of unity $\xi$, and $n$ two-dimensional irreducible representationswhere ${\omega }^3=1$ and $\omega \ne 1$. The two-dimensional representations parametrized by $\epsilon$ and $-\epsilon$ are isomorphic. The sum of squares of irreducible degrees is $2n+4n=6n$, so this list is complete.

The conjugacy classes of $G_{6n}$ arefor $0\le r<n$. With $0\le j<2n$ and $0\le k<n$, the linear character ${\lambda }_j$ and two-dimensional character ${\chi }_k$ have values