The genus 3 Chen-Gackstatter surface exists with higher dihedral symmetry, becoming symmetric with respect to reflections at vertical planes making angle π/k in addition to rotational symmetries about horizontal lines.
For given n, the resulting surface has genus 3k, and the existence proof requires to solve a 2-dimensional period problem.
Can one add vertical handles to these surfaces? This would be analogous to the k-Noids with handles and the single periodic Scherk surfaces with handles, which exist beginning with k=3.
Minimal Surfaces 