Classification of regular balanced Cayley maps of minimal non-abelian metacyclic groups

In this paper, we classify the regular balanced Cayley maps of minimal non-abelian metacyclic groups. Besides the quaternion group Q 8 , there are two infinite families of such groups which are denoted by M p ,  q ( m ,  r ) and M p ( n ,  m ) , respectively. Firstly, we prove that there are regular balanced Cayley maps of M p ,  q ( m ,  r ) if and only if q  = 2 and we list all of them up to isomorphism. Secondly, we prove that there are regular balanced Cayley maps of M p ( n ,  m ) if and only if p  = 2 and n  =  m or n  =  m  + 1 and there is exactly one such map up to isomorphism in either case. Finally, as a corollary, we prove that any metacyclic p -group for odd prime number p does not have regular balanced Cayley maps.