Example of group with normal subgroup $N\ne\{e\}$ such that $N \cap Z(G)= \{ e\} $ and $G $ \ $ N$ contains an element of order more than $2$

Question

(i) Give example of a group (if exists ) which has a normal subgroup $N\ne\{e\}$ such that

$N \cap Z(G)= \{ e\} $ and $G $ \ $ N$ contains an element of order more than $2$

(ii) Give example of a group (if exists ) such that for every normal subgroup $N\ne\{e\}$ of $G$ ,

$N \cap Z(G)= \{ e\} $ and $G $ \ $ N$ contains an element of order more than $2$

Please Help , thanks in advance .

Answer 1

Let $G=S_3\times S_3$ then $Z(G)=e\times e$ and take $N=A_3\times e$ then you are done.