It is well-known that there is an isomorphism between $PSp(4,3)$ (the symplectic group of dimension $4$ over $\mathbb F_3$) and $PSU(4,2^2)$ (the unitary group defined by $4\times4$ unitary matrices over $\mathbb F_4$).

**Question:** Can the exceptional isomorphism be interpreted in (finite) geometry?

**Known examples of geometric interpretation**:

The isomorphism $PSL(3,2)=PSL(2,7)$ is presented here.

The isomorphism $S_6=PSp(4,2)$ is explained in John Baez's article.

The isomorphism $A_6=PSL(2,9)$ can be explained by the Taylor graph of $PSL(2,9)$ acting on 10 points (i.e. the projective line of $\mathbb F_9$): this is a distance-regular graph on 20 vertices with intersection array $\{9,4,1;1,4,9\}$. Observing that this is also the intersection array of $J(6,3)$ and Johnson graphs are unique, it lets $S_6$ acts on the graph.

The isomorphism $PSU(3,3)=G_2(2)'$ can be explained by the unitary nonisotropics graph of $PΓU(3,q)$: for $q=3$, the intersection array is $\{6,4,4;1,1,3\}$, which is exactly that of a generalized hexagon of order 2. Furthermore, $GH(2,2)$ is unique up to duality, so it's also the Cayley generalized hexagon where $G_2(2)$ acts.