 Solution

In the abelianization, the relation $a^{-1}ba=b^{-1}$ becomes $b=b^{-1}$. Together with $b^3=1$ this forces $b=1$, soConsequently there are $2n$ linear characterswhere $\xi =e^{2\pi i/(2n)}$.

The representations from part (i), denoted ${\rho }_{\epsilon }$, are irreducible: the two eigenspaces of $B$ are its only invariant lines, and $A$ exchanges them. The character of a representation agrees for two parameters exactly when their squares agree, so ${\rho }_{\epsilon }\cong {\rho }_{-\epsilon }$ and there are $n$ pairwise nonisomorphic two-dimensional representations. Their squared dimensions sum toThe sum of squares of irreducible degrees proves that no others exist. Hence the complete list isThis is the classification of the Irreducible representations of G6n.