Consider the group homomorphism $\varphi : SL_2 (\Bbb Z) \longrightarrow SL_2 (\Bbb Z/ 3 \Bbb Z)$ defined by $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} \overline {a} & \overline {b} \\ \overline {c} & \overline {d} \end{pmatrix}.$$
What is the cardinality of the image of $\varphi$?
Since $\text {Im} (\varphi)$ is a subgroup of $SL_2 (\Bbb Z/ 3 \Bbb Z)$ so by Lagrange's theorem $\#\ \text {Im} (\varphi)\ \big |\ \#\ SL_2 (\Bbb Z/ 3 \Bbb Z) = 24.$ What I have observed is that $\#\ \text {Im} (\varphi) \geq 7.$ So $\#\ \text {Im} (\varphi) = 8,12\ \text {or}\ 24.$
I have just seen that the image contains at least $10$ elements. So the possibility for cardinality of $\text {Im} (\varphi)$ is $12$ or $24.$
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