What are the finite subgroups of $GL_2(\mathbb{Z})$?

What are the finite subgroups of $GL_2(\mathbb{Z})$?

I would quite like to know what the matrices which generate the subgroups are.

I know that this group has an index two subgroup which is isomorphic to $\langle x, y; x^6, y^4, x^3=y^2\rangle$ but

  • a) I cannot remember facts about free products with amalgamation for more than $5$ minutes
  • b) I do not know which matrix the $x$-generator corresponds to (or, I suppose, the $y$-generator, but I can easily find elements of order $4$…$6$ is more elusive).

Note: I have edited this answer as I got mixed up with $SL$ and $GL$ for some reason…

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