Generators and commutator Subgroup of $SL(2,3)$

Question

I'm studying about the special linear group and having some problem about the group $G=SL(2,3)$ (the same denote with $SL(2,\mathbb{F}_3)$:

1. How can i compuse the generators $\{X,Y\}$ of $G$ with the conditions $X^3=Y^3$ and $(XY)^2=Y^3$? I tried directly with some matrixs but it's not satisfy the all conditions.

(They also give a hint that $Z(G)=<Y^3>$ so $Y^3=-1$.)

2. Show that the commutator $[G,G]$ have $8$ element.

I read some documents and saw that $[G,G]$ is the $2$-Sylow subgroup of $G$ and clearly know of the 8 elements, but is there some way to compuse there element by definition?