It is [well-known](http://brauer.maths.qmul.ac.uk/Atlas/v3/clas/U42) 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](https://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7/1450#1450).


The isomorphism $S\_6=PSp(4,2)$ is explained in [John Baez's article](http://math.ucr.edu/home/baez/six.html#3).


The isomorphism $A\_6=PSL(2,9)$ can be explained by the [Taylor graph](http://mathworld.wolfram.com/TaylorGraph.html) 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 [$J(6,3)$](https://en.wikipedia.org/wiki/Johnson_graph#Intersection_array) 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](https://www.distanceregular.org/graphs/pgammau3.4.html) 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](https://www.distanceregular.org/graphs/point-gh2.2.html). Furthermore, $GH(2,2)$ is unique up to duality, so it's also the [Cayley generalized hexagon](https://en.wikipedia.org/wiki/Tutte_12-cage) where $G\_2(2)$ acts.



Geometric interpretation of the exceptional isomorphism $PSp(4,3)=PSU(4,2^2)$
I believe that a finite geometric proof is given by Jean Dieudonné here:


*Dieudonné, Jean*, [**Les isomorphismes exceptionnels entre les groupes classiques finis**](http://dx.doi.org/10.4153/CJM-1954-029-0), Can. J. Math. 6, 305-315 (1954). [ZBL0055.01904](https://zbmath.org/?q=an:0055.01904).


If you go to Section 7 of the given paper you will see the proof. Dieudonné displays the required isomorphism using a 1-1 correspondence between the 40 lines in $(\mathbb{F}\_3)^4$, and the non-isotropic lines in $(\mathbb{F}\_4)^4$ (non-isotropic with respect to a non-degenerate hermitian form).


A google-books version is [here](https://books.google.co.uk/books?id=3lYmuOh5j0YC&pg=PA303&lpg=PA303&dq=Dieudonne%20Les%20isomorphisms%20exceptionnelles&source=bl&ots=2Pg9beAb-Z&sig=ACfU3U2bGQ_gBZdJn3Tu2fL4xzVwFBNLVQ&hl=en&sa=X&ved=2ahUKEwjZmuXJrK3mAhUEQhUIHVSRB0QQ6AEwDHoECAkQAQ#v=onepage&q=Dieudonne%20Les%20isomorphisms%20exceptionnelles&f=false).


