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I am interested in knowing the semi-direct product of Quaternion group $Q_8$ with $c_p$, i.e. cyclic group of order $p$ where $p$ is a odd prime.

We know that $\text{SL}_{2}(\mathbb{Z}_{3})$ is a semi-direct product of $Q_8$ and $C_3$ from here. Semi-direct products.

Can we generalize above result to $c_p$ for odd prime $p$. Please help.

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