# External Direct Product

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.

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