  We introduce (co)homology theory for multiple group racks and construct
cocycle invariants of compact oriented surfaces in the 3-sphere using their
2-cocycles, where a multiple group rack is a rack consisting of a disjoint
union of groups. We provide a method to find new rack cocycles and multiple
group rack cocycles from existing rack cocycles and quandle cocycles.
Evaluating cocycle invariants, we determine several symmetry types, caused by
chirality and invertibility, of compact oriented surfaces in the 3-sphere.
