You can find plenty of examples by considering surfaces *isogenous to a product*, namely of the form $S=(C \times F)/G$, where $G$ is a finite group acting faithfully on the smooth curves $C$, $F$ and whose diagonal action on the product is free.

For an explicit situation, you can look at my paper "[On surfaces of general type with $p_g=q=1$ isogenous to a product of curves][1]", Corollary 2.5.


  [1]: https://arxiv.org/pdf/math/0601063.pdf