 Solution

The conjugacy classes are $\{1\}$, $\{a^n\}$, $\{a^r,a^{-r}\}$ for $1\le r\le n-1$, and two classes containing respectively $b{a}^{2s}$ and $b{a}^{2s+1}$, each of size $n$. Conjugation by $b$ inverts powers of $a$, while conjugation by $a$ changes the exponent of $b{a}^r$ by two; these facts prove the list and sizes. The sizes total $4n$.

The complete character table is specified by the following array, with one column for every $r$ and one row for every $j$ in the indicated ranges:Here $1\le r,j\le n-1$; the four pairs $(\alpha ,\beta )$ are those in part (ii). In particular, this formula displays both parity cases without concealing the different values $\pm i$ when $n$ is odd. The zero entries follow because the two-dimensional matrices representing $b{a}^r$ are off-diagonal.