Sandpile group on the graph Dn of the dihedral group

In this paper we study the structure of the Abelian sandpile group on the Cayley graph Dn of the dihedral group Dn=〈a,b∣an=b2=(ab)2=1〉. We prove that the Smith normal form of the sandpile group is not cyclic as one can generally expect but is always the direct product of two or three cyclic groups. We conclude by considering some particular cases.