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 .