Dear Forum,
I need to be able to obtain the action of the Projective Unitary
group PGU(n+1,q) on the vectors of the Projective Space PG(n,q^2).
[actually, I need this for q=5 and n=3, but I expect the solution
to be valid for any suitable n,q]
Here there is my question:
What is the sesquilinear form respected by the group PGU(n+1,q),
assuming that the vectors of PG(3,q^2) are enumerated in the order
given by
Orbit(GL(4,q^2),Z(q)^0*[1,0,0,0],OnLines) ?
[or, conversely, in what order may I enumerate the elements of
PG(n,q^2) if I wish PGU(n+1,q) to be the group preserving the
sesquilinear form induced by the identity matrix?]

Thanks in advance.
lg