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There is a semi direct product if you go $\theta: Q \rightarrow Aut(H)$.

$H = C_{17}, Q = C_2$.

$Aut(H)\cong C_{16}$.

From here, how do I construct the semi direct products?

I said, in $C_{16}$, there are two elements of order that divide $2$, which are $1$ and $2$. Therefore there are two semi direct products. Also $17 \times 2 = 34$ tells me that in both semi direct products there will be $34$ elements.

How do I construct the semi direct products from here though?

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    You know that there at most two groups of order $34$, and you know that the dihedral group of order $34$ and $C_{34}$ are groups of order $34$. Hence, it must be those two!2012-12-12

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