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 Corollary 2.5 of my paper 

*[On surfaces of general type with $p_g=q=1$ isogenous to a product of curves][1]*, Communications in Algebra **36** (2008), no. 6, 2023-2053, [arXiv:math/0601063][2].


  [1]: https://www.tandfonline.com/doi/abs/10.1080/00927870801948676
  [2]: https://arxiv.org/pdf/math/0601063.pdf