Your representation $p$ is not faithful, since we have $$ (ABA^{-1}BA^{-1}BAB^{-1})^3 \ = \ 1. $$ In particular, this means that $$ (aba^{-1}ba^{-1}bab^{-1})^3 \ = \ \left(\begin{array}{rr}% -24587&42408\\% 15048&-25955\\% \end{array}\right) $$ lies in the kernel of $p$.
Stefan Kohl ♦
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