What exactly is the group Omega(n,q) in MAGMA?

Question

Let $n>2$ and $q$ be a prime power. In MAGMA I'm having a lot of trouble identifying the group Omega(n,q). I'm trying to use a source that asserts that it is the group of $n\times n$ orthogonal matrices that have spinor norm a square in $GF(q)$, but that isn't right because when (say) $n=5$ and $q=5$ I find plenty of matrices which aren't orthogonal, such as this one: \ \begin{bmatrix} 3&3&4&3&4\\ 0&2&0&0&3\\ 2&1&2&2&3\\ 3&0&0&1&0\\ 2&3&3&2&4\\ \end{bmatrix} which we can clearly see is not orthogonal (not all columns are of length 1 and the scalar product of some columns are not 0, modulo 5). Even so, MAGMA thinks of this group as a subgroup of the GeneralOrthogonalGroup(n,q), which must have an entirely different meaning to what I thought it did. I'm trying to use the result that the projective symplectic group $PSp(4,q)$ is isomorphic to the group in question but I seem to be struggling to get a handle on what this group actually is.

Thanks in advance for the help

Answer 1

The explanation is that the symmetric bilinear form that is preserved by this group is not what you are expecting.

   > G := Omega(5,5);
   > SymmetricBilinearForm(G);
     true 
     [0 0 0 0 1]
     [0 0 0 1 0]
     [0 0 3 0 0]
     [0 1 0 0 0]
     [1 0 0 0 0]
     orthogonalcircle
   > 
   > flag, form :=  SymmetricBilinearForm(G);
   > g := Random(G);
   > g * form * Transpose(g) eq form;
     true
   > newform := IdentityMatrix(BaseRing(G),Degree(G));
   > g := TransformForm(newform, "orthogonalcircle");
   > GG := G^(g^-1);
   > SymmetricBilinearForm(GG);
     true 
     [1 0 0 0 0]
     [0 1 0 0 0]
     [0 0 1 0 0]
     [0 0 0 1 0]
     [0 0 0 0 1]
     orthogonalcircle