A homework question that I'm trying to solve is
Determine the order of $(\mathbb{Z}_{12} \times \mathbb{Z}_4)/\langle ([3], [2]) \rangle$.
I recognize that that I need to find the order of the direct product $\mathbb{Z}_{12} \times \mathbb{Z}_4$, which is $\vert \mathbb{Z}_{12} \vert \cdot \vert \mathbb{Z}_4 \vert = 12 \cdot 4 = 48$, and that I also need $\vert \langle ([3], [2]) \rangle \vert$. The thing that I'm confused about is what $ ([3],[2]) $ means in this context. I know what $[3]$ and $[2]$ mean individually, that they're congruence classes, but I don't know what $([3], [2])$ means.
Elements of $\mathbb{Z}_{12} \times \mathbb{Z}_4$ are ordered pairs $(a,b)$ where $a \in \mathbb{Z}_{12}$ and $b \in \mathbb{Z}_4$. Does that clarify things? To calculate $\vert \langle ([3], [2]) \rangle \vert$ you'll also need to know that the the product of two ordered pairs is the ordered pair of the products in the components: $$(a,b)(x,y) = (ax,by)$$