Classification of groups

1. Apr 29, 2013

Artusartos

Checking to see whether semidirect products are isomorphic.

1. The problem statement, all variables and given/known data

I want to simplify this semidrect product $(Z_7 \rtimes_{\bar{\alpha}} Z_3) \rtimes_{\alpha} Z_2$, but I'm not sure how. In other words, I want to see if this is isomorphic to (for example) $Z_7 \rtimes_{\alpha} Z_6$.

2. Relevant equations

3. The attempt at a solution

I know that $$Z_7 \rtimes_{\bar{\alpha}} Z_3$$ corresponds to the homomorphism $$\bar{\alpha}: Z_3 \rightarrow Z^{\times}_7$$, but what homomorphism does $$(Z_7 \rtimes_{\bar{\alpha}} Z_3) \rtimes_{\alpha} Z_2$$ correspond to? I need to know what $$Aut(Z_7 \rtimes_{\bar{\alpha}} Z_3)$$ is in order to look at $$\alpha: Z_2 \rightarrow Aut(Z_7 \rtimes_{\bar{\alpha}} Z_3)$$, right? But I'm not sure how to do that...