Magma V2.14-14 Fri Jan 02 2009 20:22:59 on havas-xps [Seed = 3453579159] Type ? for help. Type -D to quit. Construct Khukhro's 2-generator 7-group counterexample to the Hughes conjecture, and then repeatedly factor out complements to [b,a]^7 in the centre of the group, so as to obtain smaller counterexamples B(2,7 : 13) has order 7^668 The p-covering group G has order 7^1258 and class 14 G is generated by a and b; and [b,a] has order 49 [b,a,a] has order 7 ; [b,a,b] has order 7 gamma_3(G) is the normal closure of < [b,a,a], [b,a,b] >, and gamma_3(G) has class at most 4, so gamma_3(G) has exponent 7 Now compute suitable 7th powers of elements outside the derived group Loading "twB27c8" The quotient group H has order 7^1075 H is generated by a and b; and [b,a] has order 49 H is Khukhro's 2-generator anti-Hughes 7-group Now factor out a complement for [b,a]^7 in the multiplier Factoring out a complement of H.846 gives us a smaller anti-Hughes group CurrentQ with order 7^669 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.669^6 Z, its centre, has order 7^259 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^411 generated by a and b; and [b,a]^7 = NextQ.411^6 Z, its centre, has order 7^156 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^256 generated by a and b; and [b,a]^7 = NextQ.256^6 Z, its centre, has order 7^87 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^170 generated by a and b; and [b,a]^7 = NextQ.170^6 Z, its centre, has order 7^46 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^125 generated by a and b; and [b,a]^7 = NextQ.125^6 Z, its centre, has order 7^9 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^117 generated by a and b; and [b,a]^7 = NextQ.117^6 The centre is generated by CurrentQ.117 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^117 which is as far as we can reduce the group (by this method!) Total time: 151.468 seconds, Total memory usage: 734.27MB