Consider $f:\mathbb{Z}^3 \rightarrow \mathbb{Z}^4$ $$f(a, b, c) = (a+2b+8c, 2a-2b+4c, -2b+12c, 2a -4b + 4c)$$
Describe the image of this homomorphism as an abstract abelian group. Describe the quotient of $\mathbb{Z}^4$ by the image of this homomorphism as an abstract abelian group.
I first put the kernel into Smith normal form which has entries 1, 2, 12.
I think that this means that the image is $C_2 X C_{12}$? How do I do the last part of the question?