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Let $N$ be normal subgroup of $G$ and $G=(N\times C_{3})\rtimes C_{2}$. Then prove $G=N\times (C_{3}\rtimes C_{2})$. Thank you

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    May I suggest you this topic?http://math.stackexchange.com/questions/264096/semi-direct-groups-isomorphisms2012-12-26

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This is false. Let $N$ be any group for which $\text{Aut}(N)$ is even and let $C_2$ act trivially on $C_3$ and faithfully on $N$.

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    As a specific example we could take $N$ to be the cyclic group of order 5, and $G$ to be the direct product of the dihedral group of order 10 with a cyclic group of order 3.2012-12-26