Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 19I / b / iii / Solution

Every element has a unique form $a^s{b}^t$. Since $a^2$ is central, the conjugacy classes, indexed by $0\le r<n$, areThese have sizes one, two, and three and account for all $6n$ elements.

Take ${\rho }_k={\rho }_{{\xi }^k}$ for $0\le k<n$. On an even power of $a$, its representing matrix is scalar; on an odd power it is off-diagonal. Since $\omega +{\omega }^2=-1$, the full Character table of G6n isfor $0\le j<2n$, $0\le k<n$, and $0\le r<n$.