How one can find the abelian group which has a presentation $$\langle x,y,z,w\mid6x+8y+10z+14w, 4x+4y+4z+4w\rangle$$ Is there any way indicates the steps to find such a group? Or just by guesswork and experience?
Edit: Can one proves directly whether it is the group $\mathbb Z_2 \times \mathbb Z \times \mathbb Z \times \mathbb Z$ or not?
The smith normal form of the matrix $$\begin{pmatrix}2 & 4 & 6 & 8 \\ 4 & 4 & 4 & 4\end{pmatrix}$$ is $$\begin{pmatrix}2 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0\end{pmatrix},$$ so your group is isomorphic to $$\mathbb Z/2\mathbb Z \times \mathbb Z/4\mathbb Z\times\mathbb Z\times\mathbb Z.$$