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Let $S_n$ denote the symmetric group on $n$ symbols. The group $S_3\oplus(\Bbb Z/2\Bbb Z)$ is isomorphic to which of the following groups?

1.$\Bbb Z/12\Bbb Z$

2.$\Bbb Z/6\Bbb Z \oplus \Bbb Z/2\Bbb Z$

3.$ A_4$, the alternating group of order $12$

4.$ D_6$ the dihedral group of order $12$.

I can easily discard option 1st and 2nd. Since $S_3$ is non abelian implies $S_3\oplus(\Bbb Z/2\Bbb Z)$ s non abelian. But I have problem to deal with other options. Will be pleased if you share your valuable knowledge.

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Since $((123), [1]_2)$ has order six and $A_4$ has no element of order six, the answer must be 4., the dihedral group $D_6$.

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