We study the properties of the surface
Σ
q
\Sigma _q
, which is a
2
q
2q
-fold cover of
H
/
G
q
\mathbb H/G_q
, where
G
q
G_q
is a Hecke group and
q
q
is an integer greater than
3
3
. We have slightly different situations for the even and odd values of
q
q
. For odd values of
q
q
the surface
Σ
q
\Sigma _q
is a
q
−
1
2
\frac {q-1}{2}
genus surface with a cusp, whereas, for even values it is a
q
−
2
2
\frac {q-2}{2}
genus surface with two cusps. We prove that there exist
g
g
embedded tori with a hole on
Σ
q
\Sigma _q
, where
g
=
q
−
1
2
g=\frac {q-1}{2}
when
q
q
is an odd integer and
g
=
q
−
2
2
g=\frac {q-2}{2}
when
q
q
is even, with
g
g
boundary geodesics at different heights. These boundary geodesics are the separating geodesics intersecting each other transversally. We also prove that the surface
Σ
q
\Sigma _q
is a hyper-elliptic surface for every integer
q
>
3
q>3
.