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Let $G$ be the group with this presentation $$G=\left\langle x,\,y\,|\,x^n,\,\,y^m,\,\,xyx^{n-1}y\right\rangle $$ I know that if $n$ is even is a semidirect product of the kind $\mathbb Z_m \rtimes \mathbb Z_n$ and I know that if $n=2$ is the dihedral group $D_m$. But does it have a specific name for others $n$?

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  • $\begingroup$ I don't know of any name. But note that $Z(G) = \langle x^2 \rangle$ and $G/Z(G) \cong D_m$. $\endgroup$ – Derek Holt May 23 '17 at 14:10
  • $\begingroup$ Thank you Derek I didn't notice that $\endgroup$ – Dac0 May 23 '17 at 14:48
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I did a bit of looking, but I suspect they do not. Remember that a semidirect product $G\rtimes_\varphi\mkern-3mu H$ is completely determined by the normal subgroup $G$, the subgroup $H$ and the homomorphism $\varphi : H \to \text{Aut}(G)$, so in some sense a name for these groups would be superfluous.

Dihedral as a descriptor refers to the fact that these groups (along with the infinite group $\mathbb Z \rtimes \mathbb Z_2 = \langle a,b \mid b^2, bab=a^{-1}\rangle$) arise as the group generated by reflections across two planes as the (dihedral) angle between them varies. If some of the semidirect products you mention have other names, it will likely be because of a similar connection to another area of math.

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  • $\begingroup$ Thank you... let's see if someone has something else, or some other special case which deserves a name $\endgroup$ – Dac0 May 23 '17 at 14:46

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