Regular Cayley maps on dihedral groups with the smallest kernel

Abstract

Let $$\mathcal {M}={{\mathrm{CM}}}(D_n,X,p)$$M=CM(Dn,X,p) be a regular Cayley map on the dihedral group $$D_n$$Dn of order $$2n, n \ge 2,$$2n,nź2, and let $$\psi $$ź be the skew-morphism associated with $$\mathcal {M}$$M. In this paper it is shown that the kernel $${{\mathrm{Ker}}}\psi $$Kerź of the skew-morphism $$\psi $$ź is a dihedral subgroup of $$D_n$$Dn and if $$n \ne 3,$$nź3, then the kernel $${{\mathrm{Ker}}}\psi $$Kerź is of order at least 4. Moreover, all $$\mathcal {M}$$M are classified for which $${{\mathrm{Ker}}}\psi $$Kerź is of order 4. In particular, besides four sporadic maps on 4, 4, 8 and 12 vertices, respectively, two infinite families of non-t-balanced Cayley maps on $$D_n$$Dn are obtained.