I have the following problem: I have to describe up to isomorphism the semidirect product $C_6 \rtimes C_2$, where $C_6$ denotes cyclic group of order six. I think I have to use external semidirect products.
So first I search for group $AutC_6$ and I find 2 automorphisms:
- $\sigma_0 = id_{C_6}$,
- $\sigma_1 = [x \rightarrow x^5]$.
To describe all semidirect products $C_6 \rtimes_{\phi} C_2$ I find all homomorphisms $\phi: C_2 \rightarrow AutC_6$. I look only at generator in $C_2 = <a>$.
First of all, I see that I can set $\phi(a) = \sigma_0$, so my $\phi$ sends all to identity. That $\phi$ is a proper homomorphism so by that I have my first external semidirect product.
Then I set $\phi(a) = \sigma_1$. I see that such $\phi$ is a proper homomorphism since
$id = \phi(e) = \phi(a a) = \phi(a) \circ \phi(a) = [x \rightarrow x^{25}] = [x\rightarrow x]$,
as we are in $C_6$.
Am I doing everything all right? Does that mean that there are two semidirect product? What should I do with the 'up to isomorphism part'?
Thanks in advance