Regular Cayley maps on dihedral groups with the smallest kernel

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Let M=CM(Dₙ,X,p) be a regular Cayley map on the dihedral group Dₙ of order 2n, n ≥ 2, and let π be the power function associated with M. In this paper it is shown that the kernel Ker(π) of the power function π is a dihedral subgroup of Dₙ and if n ≠ 3, then the kernel Ker(π) is of order at least 4. Moreover, all M are classified for which Ker(π) is of order 4. In particular, besides 4 sporadic maps on 4,4,8 and 12 vertices respectively, two infinite families of non-t-balanced Cayley maps on Dₙ are obtained.