I was playing with the Hermitian curve y^q + y = x^{q+1} over the field GF(q^2) and chanced upon the following (non Abelian) group law on the points of the affine curve:

(a,b) * (c,d) = (a+c,b+d+ac^q)

Over GF(q^2) the group has q^3 points, the identity is (0,0) and the inverse of (a,b) is (-a,b^q).

So my question is whether this is known. I could not find a reference. If you haven't seen this before and if you're interested in seeing how I found this group law, just shoot me an email.

Thanks for your time. Best, Hiren