I know that the direct product operation is associative, and that in general the semidirect product operation is not associative. But what about when we work with a mix of both?
I'm trying to identify the isomorphism class of the following group, and I'm guessing it's either $D_{12}$, the dihedral group of order $12$ or $\Gamma_{12}$, the dicyclic group of order $12$: $$C_3 \rtimes V_4 \cong C_3 \rtimes (C_2 \times C_2)$$ The homomorphism defining the semidirect product is the non-trivial one. I strongly believe that it is isomorphic to $\Gamma_{12}$, but I'm not beign able to show why. It would be pretty straightfoward if I could associate the products like this: $$C_3 \rtimes (C_2 \times C_2)$$ $$\cong C_3 \rtimes C_2 \times C_2$$ $$\cong (C_3 \rtimes C_2) \times C_2$$ $$\cong D_6 \times C_2 \cong \Gamma_{12}$$ But I'm not confortable with doing it, because I have no arguments to believe it's correct. Is there some general "associative" property with mixed products I could use? Or it needs to be shown by another way, like homomorphisms between the products that makes this association valid in this case? Also, it could be isomorphic to something else and I'm wrong from the start, and in that case I gladly accept some light.
EDIT: Another group I'm classifying falls in the same associativity problem: $V_4 \rtimes C_3$. Could it be $C_2 \times C_2 \rtimes C_3 \cong C_6 \times C_2$?