4. (i) Consider $G=\mathbb{Z}^{3}$ be a free abelian group generated by $u_{1}$, $u_{2}$, and $u_{3}$, and let $H$ be the subgroup generated by $$ 2 u_{1}+12 u_{3}, \quad 2 u_{2}-2 u_{3}, \quad 2 u_{1}+4 u_{2} . $$ (a) Determine generators $v_{1}, v_{2}$, and $v_{3}$ for $G$ such that $H$ is generated by $h_{1} v_{1}, h_{2} v_{2}, h_{3} v_{3}$ with $h_{1}\left|h_{2} ight| h_{3}$. [6 Marks ] (b) Describe $G / H$ as a product (direct sum) of cyclic groups. [4 Marks] (ii) Give the full list of all non-isomorphic abelian groups of order $294 $ write their elementary divisors and invariant factors. [4 Marks] CS.VS. 1596