/*
Group G = 2^4:L3(2) x 3 x 7 = 2^{1.3}:L3(2) x 21.
Automorphism group is 2^3:L3(2) x 2 x 6, and has order \phi(|G|).
Should work in Magma 2.8 and above.
|G|     = 56448 = 2^7 * 3^2 * 7^2.
|Aut G| = 16128 = 2^8 * 3^2 * 7.
Second smallest known non-cyclic group for which |Aut G| = \phi(|G|).
*/

G<g1,g2,g3>:=PermutationGroup<26|
(1,3)(2,4)(5,15,6,16)(7,9)(8,10)(11,13,12,14),
(1,3,5)(2,4,6)(7,15,11)(8,16,12),
(17,18,19)(20,21,22,23,24,25,26)>;

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