Regular Cayley maps on dihedral groups with the smallest kernel

Let $\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group $D_n$ of order $2n, n \ge 2,$ and let $π$ be the power function associated with $\mathcal{M}$. In this paper it is shown that the kernel Ker$(π)$ of the power function $π$ is a dihedral subgroup of $D_n$ and if $n \ne 3,$ then the kernel Ker$(π)$ is of order at least $4$. Moreover, all $\mathcal{M}$ 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 $D_n$ are obtained.

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