/*
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<x,y,z>:=PermutationGroup<346|
(2,3,5,8)(4,7,12,20)(6,10,17,27)(9,15,16,26)(11,19,31,47)(13,21,32,49)
(14,23,36,57)(18,29,45,72)(22,35,55,86)(24,38,60,93)(25,39,62,97)(28,43,69,70)
(30,46,74,114)(33,51,80,122)(34,53,83,127)(37,58,90,136)(40,64,100,149)
(41,65,102,152)(42,67,105,157)(44,71,109,162)(48,77,119,172)(50,79,75,115)
(52,82,125,180)(54,85,129,185)(56,88,132,190)(59,92,139,198)(61,95,126,135)
(63,99,147,210)(66,104,156,84)(68,106,159,221)(73,112,164,227)(76,117,169,146)
(78,120,173,174)(81,124,178,242)(87,131,189,251)(89,134,193,195)
(91,138,197,260)(94,142,203,266)(96,144,207,111)(98,145,209,186)
(103,154,217,108)(107,160,208,271)(110,151,215,277)(113,166,230,264)
(116,168,140,200)(118,171,236,153)(121,175,239,212)(123,176,234,272)
(128,184,249,165)(130,187,188,250)(133,192,137,196)(141,150,213,274)
(143,205,268,238)(148,163,226,293)(155,181,158,219)(161,224,191,254)
(167,231,297,333)(170,235,301,336)(177,241,232,298)(179,244,309,261)
(182,247,312,327)(183,248,211,273)(194,256,280,331)(199,263,319,342)
(201,204,246,311)(202,265,321,322)(206,270,318,294)(214,276,326,323)
(216,279,329,228)(218,281,332,302)(220,284,289,306)(222,286,257,316)
(223,288,258,317)(225,291,335,262)(229,296,328,340)(233,299,245,310)
(237,303,338,253)(240,304,269,325)(243,307,339,259)(252,267,283,314)
(255,315,292,305)(275,295,300,337)(278,324,341,343)(287,334,330,313)
,
(1,2,4)(3,6,11)(5,9,16)(7,13,22)(8,14,24)(10,18,30)(15,25,40)(17,28,44)
(19,32,50)(20,33,52)(21,34,54)(23,37,59)(26,41,66)(27,42,68)(29,31,48)
(35,56,36)(38,61,96)(39,63,86)(43,70,108)(45,73,113)(46,75,116)(47,76,118)
(49,78,121)(51,81,122)(53,84,119)(55,87,67)(57,89,135)(58,91,74)(60,94,131)
(62,98,146)(64,101,151)(65,103,155)(69,107,161)(71,110,163)(72,111,97)
(77,117,170)(79,83,128)(80,123,177)(82,126,182)(85,130,188)(88,133,138)
(90,137,115)(92,140,201)(93,141,202)(95,143,206)(99,148,212)(100,150,214)
(102,153,216)(104,132,191)(105,158,220)(109,147,211)(112,165,229)(114,167,232)
(120,174,231)(124,179,175)(125,181,246)(127,183,203)(129,186,193)(134,194,257)
(136,195,258)(139,199,173)(142,204,267)(144,208,251)(145,190,253)(152,197,261)
(154,218,282)(156,205,269)(157,198,262)(159,222,287)(160,223,289)(162,225,292)
(164,228,271)(166,207,210)(168,233,300)(169,234,291)(171,237,301)(172,238,299)
(176,240,305)(178,243,308)(180,245,270)(184,248,313)(185,224,290)(187,196,259)
(189,252,297)(192,255,273)(200,264,298)(209,272,288)(213,275,242)(215,278,328)
(217,280,230)(219,283,236)(221,285,333)(226,294,336)(227,295,331)(235,302,321)
(239,260,318)(241,306,281)(244,279,330)(247,307,256)(249,268,324)(250,314,341)
(263,320,304)(265,322,325)(266,323,334)(274,310,340)(276,327,293)(277,316,286)
(284,309,312)(296,329,317)(303,319,343)(311,335,337)(315,338,326)
,
(344,345,346)
>;

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