Cyclic and dihedral NNN-groups and NNND-groups

Abstract A Cayley digraph on a group 𝐺 is called NNN if the Cayley digraph is normal and its automorphism group contains a non-normal regular subgroup isomorphic to 𝐺. A group is called an NNND-group or an NNN-group if there is an NNN Cayley digraph or graph on the group, respectively. In this paper, it is shown that there is no cyclic NNND-group, and hence no cyclic NNN-group. Furthermore, a dihedral group of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> 2n is an NNND-group or an NNN-group if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>6</m:mn> </m:mrow> </m:math> n\geq 6 is even and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≠</m:mo> <m:mn>8</m:mn> </m:mrow> </m:math> n\neq 8 .