  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.
