Abstract
In this paper, we classify the regular balanced Cayley maps of minimal non-abelian metacyclic groups. Besides the quaternion group Q8, there are two infinite families of such groups which are denoted by Mp, q(m, r) and Mp(n, m), respectively. Firstly, we prove that there are regular balanced Cayley maps of Mp, 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 Mp(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.
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