Semidirect product of Heisenberg group 3 and $SL(2,3)$

Question

I've been trying to build the semidirect of $He_3$ and $SL(2,3)$ in Magma, but haven't had much luck yet. This may be because of a lack of experience with the software, but any help would be amazing. I'm also trying to build the semidirect products of $He_q$ and $SL(2,q)$ for $q=5,7$, but I assume such constructions are similar for the case in which $q=3$. Thank you!

Answer 1

I think the function below returns what you want as a permutation group of degree $p^3$. (I don't think there is a faithful permutation representation of smaller degree.)

SDH := function(p)
  E := ExtraSpecialGroup(p,1);
  H, phi := Holomorph(E);
  N := phi(E);  //the image of E in H
  D := DerivedGroup(H);  //to get SL on top
  P := pCore(D,p);
  C := Complements(D,P);
  return sub< D | N, C[1] >;
end function;

In fact for $p=3$, you can get the group you want more easily as $\mathtt{SmallGroup}(648,533)$, but for larger $p$ the groups are too large for the small groups library.