/*
Group G = M11 x 2 x 3 x 5 x 11.
Automorphism group is M11 x 1 x 2 x 4 x 10, and has order \phi(|G|).
Should work in Magma 2.8 and above.
|G|     = 2613600 = 2^5 * 3^3 * 5^2 * 11^2.
|Aut G| =  633600 = 2^8 * 3^2 * 5^2 * 11.
*/

G<x,y,z>:=PermutationGroup<32|
(1,11)(2,3)(5,10)(7,8),
(2,4,5,11)(6,7,9,10),
(12,13)(14,15,16)(17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32)
>;

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