The evidence of the existence of tetra-quark meson and penta-quark fermion systems has revived interest in the hunt of other multi-quark systems. We explore the possibility of six-quark cluster configurations with the two nucleon spin-isospin quantum numbers existing in the nuclear medium. The investigation of such clusters is feasible if one exploits all the symmetries involved, in particular orbital, color-spin SU (6)cs and isospin SU(2). We investigate whether such clusters contribute to neutrinoless double-beta decay mediated by heavy neutrinos or other exotic particles, since in conventional nuclear physics the relevant nuclear matrix elements are suppressed due to the presence of the nuclear hard core.