Let $G$ be a finite group with automorphism group ${\rm Aut}(G)$. Let $A_G$ denote the semidirect product of $G$ with ${\rm Aut}(G)$ in the canonical way.

Question 1: Is there are name for $A_G$ in the literature?

Question 2: Is there a way to obtain $A_G$ in GAP (and/or Magma) for a given finite group $G$?

  • $\begingroup$ In Robinson's, "A Course in the Theory of Groups (Second Edition)", it is called the holomorph of $G$ and denoted by $${\rm Hol}\, G.$$ $\endgroup$ – Shaun Apr 9 at 21:29
  • 2
    $\begingroup$ There is a Magma function $\mathtt{Holomorph}(G)$. $\endgroup$ – Derek Holt Apr 9 at 22:25

I have seen this semidirect product be called the holomorph. One can create it in GAP simply as SemidirectProduct with no extra work:

gap> g:=AlternatingGroup(6);;
gap> a:=AutomorphismGroup(g);
<group with 4 generators>
gap> h:=SemidirectProduct(a,g);
<permutation group with 6 generators>
gap> Size(h);

It's called the holomorph. For instance, for $S_n$ if $n\ne2,3,6$, we get the canonical semidirect product of $S_n$ with itself: $S_n\rtimes S_n$.


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