/*
Group G = 7^{1+2}:2S4^- x 3.
Automorphism group is 7^2:(2S4 x 3) x 2, and has order \phi(|G|).
Should work in Magma 2.8 and above.
|G|     = 49392 = 2^4 * 3^2 * 7^3.
|Aut G| = 14112 = 2^5 * 3^2 * 7^2.
Smallest known non-cyclic group for which |Aut G| = \phi(|G|).
*/

G<g1,g2,g3,g4,g5,g6,g7,g8,g9>:=PCGroup(\[9,3,2,3,2,2,2,7,7,7,442,119,1092,
777,174,1903,697,301,130,414303,34800,72609,16422,6981,259216,133081,183202,
24523,34324,4093]);

x:=g2;
y:=g3*g4*g5*g7^2*g8^3*g9^3;
z:=g1;

AU:=AutomorphismGroup(G);
#G;
#AU,"\t",EulerPhi(#G);