As I recall, the finite subgroups of $PGL_N(\mathbb C)$ are classified for $N \le 7$. See Miller--Blichfeldt--Dickson's "Theory of finite groups" (1916) for $N=3$ or Blitchfeldt's "Finite collineation groups" (1917) for $N=3, 4$. (Beware: the terminology is quite old---for instance, isomorphic only means something like isomorphic modulo a normal (or maybe central) subgroup in modern notation.)
Feit (The current situation in the theory of finite simple groups. Actes du Congr`es International des Math'ematiciens, Nice 1970) gives a list of maximal finite subgroups in these cases, though there are a few cases that he missed in higher dimensions.
Incidentally, in my thesis, I tried classifying the finite solvable subgroups of $GSp_4(\mathbb C)$ in a more simple manner than Blitchfeldt, though at one crucial point, I couldn't get around using some of his work. Still, it might be useful (and easier to read than Blitchfeldt) if you want to know some basic techniques in these classification questions.
EDIT: It turns out I wrote down the list of primitive subgroups of $PGL_3(\mathbb C)$ in my thesis. (I knew I had it written down somewhere!)
http://thesis.library.caltech.edu/2141/1/Thesis.pdf
See chapter 8 for the list: it consists of 6 groups: three nonsimple of orders 36, 72 and 216; and $A_5 \simeq PSL(2,5)$, $A_6 \simeq PSL(2,9)$ and $PSL(2,7)$. See Appendix B for the GAP notation of the first 3 groups.
RE-EDIT: The paper
Ian Hambleton and Ronnie Lee, Finite group actions on $\mathbb P^2(\mathbb C)$, J. Alg. (1988)
has a description of all finite subgroups of $PGL_3(\mathbb C)$, including the 4 classes of imprimitive groups.
Best Answer
Let $\omega$ be a primitive fifth root of $1$. By a back of the envelope calculation, using traces, I came up with
$$ x \mapsto \left( \begin{array}{cc}\omega&0\\0&\omega^{-1}\end{array}\right),\ \ \ \ x^{-1}y \mapsto \left( \begin{array}{cc}a&b\\1&d\end{array}\right), $$
where $a = 1/(\omega^2-1)$, $b=-a^2-a-1$, $d=-1-a$, and the images are of course to be understood to be modulo the scalar matrices.