= Nonnormal finite cover of a higher-genus orientable surface
For $g\geq2$ and $n\geq3$, send
$$
a_1\mapsto\sigma,\quad b_1\mapsto\tau,\quad
a_2\mapsto\tau,\quad b_2\mapsto\sigma,
$$
where $\sigma=(1\,2\,\cdots\,n)$ and $\tau=(1\,2)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.
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