Presentation of $\mathbb{Z}_9\rtimes\mathbb{Z}_3$ (Semi direct product of $\mathbb{Z}_9$ and $\mathbb{Z}_3$)

Question

I want Presentation of $\mathbb{Z}_9\rtimes\mathbb{Z}_3$ which is semi direct product of $\mathbb{Z}_9 $ and $\mathbb{Z}_3$. My first guess is as below

$$\mathbb{Z}_9\rtimes\mathbb{Z}_3= \{a,b:a^9=b^3=e, bab^{-1}=a^4\}.$$ Which i tried by using GAP just by looking at elements.

Please tell me is it right or not. Thanks.

Answer 1

Semidirect products of cyclic groups $C_n$ by $C_m$, i.e. $C_n \rtimes C_m$ have the presentation $$ \langle x,y∣x^n=y^m=1,y^{-1}xy=\phi(y)^{-1}xφ(y) \rangle, $$ where $φ:C_m→U(n)$ is the chosen homomorphism, and $y$ generates $C_m$. The proof can be found on this site.