Describing holomorph of dihedral group

Question

Are there any articles or books, where ${\rm Hol}(D_{2n}$) ($D_{2n}$ is dihedral group of order $2n$), is described?Semidirect product seems like very hard thing to compute, but I'm interested whether this holomorph is embedded in something nice.

Answer 1

So the holomorph of a group (you may know) is the semi-direct product $$\rm hol(G):=G\rtimes \rm Aut (G),$$ where it's to be understood that we have the identity homomorphism $\varphi: \rm Aut(G)\to\rm Aut(G).$

Thus it's not abelian.

In the case of $D_{2n},$ we have that $$\rm Aut(D_{2n})\cong \rm hol(\Bbb Z_n).$$

Thus we get $$\rm hol(D_{2n})\cong \rm D_{2n}\rtimes \rm hol(\Bbb Z_n).$$

But, $$\rm hol(\Bbb Z_n)\cong \Bbb Z_n\rtimes \Bbb Z_n^×.$$

So, we get $$D_{2n}\rtimes (\Bbb Z_n\rtimes \Bbb Z_n^×).$$

Thus we get that the order is $$2n^2\varphi (n)$$ (Euler's totient).

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So, it's embedded in $S_{2n^2\varphi (n)},$ by Cayley's theorem.

But better can be done. Thanks to @Derek Holt for pointing out that the holomorph naturally embedds in $S_{\lvert G\rvert},$ or $S_{2n}$ in this case.