Group Extension Example

I am trying to find two distinct group extension of $Z_2$ by $Z_3$. One "natural" extension that I found was $(Z_3,Z_2)$. That is $0 \rightarrow Z_3\rightarrow (Z_3,Z_2)\rightarrow Z_2\rightarrow 0$. However, I am having difficulty in finding another one. Any hints?

• Is $(\Bbb{Z}_3, \Bbb{Z}_2) = \Bbb{Z}_3 \times \Bbb{Z}_2$, the cartesian product? – Sammy Black Oct 13 '14 at 6:29
• Do you know about semidirect products? – Sammy Black Oct 13 '14 at 6:30
• @SammyBlack: Yes it is cartesian product. Yes, I am familliar with semidirect products – Rutherford Mark Oct 13 '14 at 6:36

There are two (split) extensions, depending on which homomorphism you choose $$\Bbb{Z}_2 \to \operatorname{Aut} \Bbb{Z}_3 \cong C_2.$$ (I prefer the cyclic group notation $C_2$ for automorphisms that are written multiplicatively, although, of course, $C_2 \cong \Bbb{Z}_2$ as abstract groups.)
If you choose the trivial homomorphism, then you get the direct product. Whereas, if you choose the isomorphism, then you get the semidirect product that is isomorphic to $D_3 \cong S_3$, the symmetries of an equilateral triangle. (The first map embeds $\Bbb{Z}_3$ as the subgroup $A_3$ of rotations and the second projects onto the sign of the permutation.)
• $\Bbb{Z}_3 \times \Bbb{Z}_2 \cong \Bbb{Z}_6$ is abelian, while $\Bbb{Z}_3 \rtimes \Bbb{Z}_2 \cong S_3$ is not. – Sammy Black Oct 13 '14 at 6:55