On a {K₄,K_(2,2,2)}-ultrahomogeneous graph

The existence of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph $G$ is established, (i.e. each isomorphism between two copies of $K_4$ or $K_{2,2,2}$ in $G$ extends to an automorphism of $G$), with the 42 ordered lines of the Fano plane taken as vertices. This graph $G$ can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of $K_4$ and as the edge-disjoint union of 21 induced copies of $K_{2,2,2}$, with no more copies of $K_4$ or $K_{2,2,2}$ existing in $G$. Moreover, each edge of $G$ is shared by exactly one copy of $K_4$ and one of $K_{2,2,2}$. While the line graphs of $d$-cubes, ($3\le d\in\ZZ$), are $\{K_d, K_{2,2}\}$-ultrahomogeneous, $G$ is not even line-graphical. In addition, the chordless 6-cycles of $G$ are seen to play an interesting role and some self-dual configurations associated to $G$ with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered.