The classification conjecture for groups admitting oriented regular representations

An oriented regular representation (ORR) of a finite group $G$ is an oriented Cayley digraph on $G$ whose automorphism group is the right-regular representation of $G$. A group is generalized dihedral if it is a generalized dihedral group in the usual sense. Let $C_n$ denote the cyclic group of order $n$, let $Q_8$ denote the quaternion group of order $8$, and let $D_4\circ D_4$ denote the central product of two dihedral groups of order $8$.

The ORR classification conjecture. Every finite group $G$ admits an ORR, unless one of the following occurs:

1. $G$ is generalized dihedral with $|G|>2$; or
2. $G$ is isomorphic to one of the eleven groups

$$Q_8,\,C_4\times C_2,\, C_4\times C_2^2,\, C_4\times C_2^3,\, C_4\times C_2^4,\,C_3^2,\,C_3\times C_2^3,$$ $$\langle a,b\mid a^4=b^4=(ab)^2=(ab^{-1})^2=1\rangle,$$

where the group has order $16$;

$$\langle a,b,c\mid a^4=b^4=c^4=(ba)^2=(ba^{-1})^2=(bc)^2=(bc^{-1})^2=1,$$ $$a^2=c^2,a^c=a^{-1}, a^2=b^2\rangle,$$

where the group has order $16$;

$$\langle a,b,c\mid a^4=b^4=c^4=(ab)^2=(ab^{-1})^2=1,$$ $$(ac)^2=(ac^{-1})^2=(bc)^2=(bc^{-1})^2=a^2b^2c^2=1\rangle,$$

where the group has order $32$; or $D_4\circ D_4$, the extraspecial group of order $32$ of plus type.

The conjecture would classify precisely the finite groups that do not admit ORRs. The paper establishes the result for all finite non-solvable groups and analyzes broad classes of groups generated by at most three elements, while the proposed classification for all finite groups remains open.