There is a unique simple group of order 25920. It has three main forms that I am aware of $$ B_2(3) \cong C_2(3) \cong \; ^2A_3(2^2) $$ equivalently $$ O(5,3) \cong PSp(4,3) \cong PSU(4,2) $$ It is well known that $ B_2(q) \cong C_2(q) $ in other words $ O(5,q)\cong PSp(4,q) $. So the first isomorphism is just this for the case $ q=3 $.
However I would like to better understand the isomorphisms $ PSp(4,3) \cong PSU(4,2) $ and $ O(5,3) \cong PSU(4,2) $. Is there any nice geometric way to see this, for example by seeing these groups as the symmetries of related objects ?
To be honest I don't even know much about the $ O(5,q) \cong PSp(4,q) $ isomorphism so if someone thinks that explaining that more or providing a helpful reference would help then I would certainly appreciate that as well.
for example something like this:
