Yesterday i encountered the following question: let $G$ be the group
$$G \ =\ \{\left(\begin{array}{cc} a & b \\ c & d \end{array}\right) \mid a,b,c,d\in \mathbb{Z}_{3},\ ad-bc\neq 0\}.$$ The question was to evaluate the order of $G$. By brute-force I figured out the order to be $48$. But it took so much time. My question is that is there any formula or rule to find the orer of such groups in general case like when all the entries of the matrix are from $\mathbb{Z}_{p}$ where $p$ is any prime.
