A finite quasigroup S is a finite algebraic system that satisfies the unique solution to the linear equations ax=b and ya=b for any a and b in S. Example of quasigroup is a group.
GAP (Groups, Algorithms, Programming) is an open source system for computational discrete algebra that is available for Windows, Linux, and OS X. See http://www.gap-system.org/ . It has quite a bit of functionality for Cayley tables of quasigroups, see http://www.gap-system.org/Manuals/pkg/loops/htm/CHAP004.htm