= Solution
The <dihedral group> here has order $2n$. Substituting $h=2$ into part (d) gives
$$
\boxed{g=2n+1.}
$$
In particular, $n=3$ requires <genus> seven. A geometric construction is to start with a round <solid torus> about the vertical axis, then drill $2n$ thin radial tunnels through its tube in the horizontal midplane, equally spaced at azimuths $k\pi/n$, $0\leq k<2n$. The tunnels run from the central opening to the exterior; their directions are horizontal and radial. Round all their rims in the same way, preserving the <dihedral symmetry>, and let $X$ be the <boundary> of the resulting solid.
\Image[/past-exam-of-the-geometry-of-surfaces-course-of-the-university-of-oxford/2023-dihedral-free-action.png]
{title=A genus-seven surface with a free dihedral action: six radial bores contain the horizontal rotation axes}
Here is why the construction has exactly the required properties. Every nonidentity power of $\beta$ is a <rotation in three dimensions> fixing the vertical axis, which lies in the central opening and is disjoint from $X$. Each element $\alpha\beta^j$ is a half-turn about a horizontal line at azimuth $j\pi/n$, modulo $\pi$. The $n$ such lines meet the original <solid torus> in $2n$ short radial segments. Each segment lies strictly inside one drilled tunnel, so these axes are also disjoint from $X$. A nonidentity <rotation in three dimensions> fixes only its axis. Thus no nonidentity element fixes a point of $X$, proving a <free action of a group>.
Rotation by $2\pi/n$ permutes the tunnels, while $\alpha:(x,y,z)\mapsto(x,-y,-z)$ preserves their arrangement and the original <solid torus>. Hence $X$ is preserved by the two generators and therefore by $G$. The tunnels can be chosen sufficiently thin to be pairwise disjoint, and the rounding can be made <group>-invariant in small disjoint <neighbourhoods> of their rims. It neither changes the <genus> nor introduces an intersection with any axis.
Each tunnel removes two discs from the existing <boundary> and inserts the cylindrical wall of the bore. Its contribution to <Euler characteristic> is $-2$: the two removed discs contribute $-2$, and the inserted <annulus> and its boundary <circles> have zero <Euler characteristic>. The <boundary> remains <connected> and <orientable>, so each tunnel increases <genus> by one. Starting from a <torus> and drilling $2n$ tunnels therefore gives $g=1+2n$. The construction also supplies a <Riemann surface>: use the <complex structure> associated with the induced oriented <Riemannian metric>. The ambient generators are <isometries> preserving the solid and ambient <orientation of a vector space>, so they preserve the induced <boundary orientation> and act by <biholomorphisms>. Part (d) then confirms that the <quotient space> has <genus> two. This is a <free dihedral action on a drilled torus>.
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