Magma V2.14-14 Fri Jan 02 2009 21:26:47 on havas-xps [Seed = 1314586363] Type ? for help. Type -D to quit. Construct a 2-generator anti-Hughes 7-group of class 14 which satisfies 6 commutator defining relators. Then repeatedly factor out complements to [b,a]^7 in the centre of the group, to obtain smaller counterexamples B(2,7 q6 : 13) has order 7^141 The p-covering group G has order 7^236 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 "twB27c8q" The quotient group H has order 7^165 H is generated by a and b; and [b,a] has order 49 H is anti-Hughes; the normal closures of a and b both have class 7 Now factor out a complement for [b,a]^7 in the multiplier Factoring out a complement of H.142 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^5 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^5 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^5 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^5 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^5 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.143 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^2 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^2 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^2 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^2 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^2 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.144 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^4 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^4 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^4 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^4 Z, its centre, has order 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^72 generated by a and b; and [b,a]^7 = NextQ.72^4 The centre is generated by CurrentQ.72 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^72 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.145 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^6 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^6 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^6 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^6 Z, its centre, has order 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^72 generated by a and b; and [b,a]^7 = NextQ.72^6 The centre is generated by CurrentQ.72 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^72 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.146 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^3 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^3 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^3 Z, its centre, has order 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^72 generated by a and b; and [b,a]^7 = NextQ.72^3 The centre is generated by CurrentQ.72 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^72 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.149 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.150 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^2 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^2 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^2 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^2 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^2 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.151 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^5 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^5 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^5 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^5 Z, its centre, has order 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^72 generated by a and b; and [b,a]^7 = NextQ.72^5 The centre is generated by CurrentQ.72 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^72 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.152 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^2 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^2 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^2 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^2 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^2 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.153 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^3 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^3 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^3 Z, its centre, has order 7^7 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^71 generated by a and b; and [b,a]^7 = NextQ.71^3 The centre is generated by CurrentQ.71 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^71 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.155 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^3 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^3 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^3 Z, its centre, has order 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^72 generated by a and b; and [b,a]^7 = NextQ.72^3 The centre is generated by CurrentQ.72 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^72 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.156 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^6 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^6 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^6 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^6 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^6 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.157 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^2 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^2 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^2 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^2 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^2 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.158 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^3 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^3 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^3 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^3 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.160 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.161 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.162 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^6 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^6 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^6 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^6 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^6 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.163 gives us a smaller anti-Hughes group CurrentQ with order 7^142 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^4 Z, its centre, has order 7^27 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^116 generated by a and b; and [b,a]^7 = NextQ.116^4 Z, its centre, has order 7^26 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^91 generated by a and b; and [b,a]^7 = NextQ.91^4 Z, its centre, has order 7^15 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^77 generated by a and b; and [b,a]^7 = NextQ.77^4 Z, its centre, has order 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^73 generated by a and b; and [b,a]^7 = NextQ.73^4 The centre is generated by CurrentQ.73 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^73 which is as far as we can reduce the group (by this method!) Total time: 14.515 seconds, Total memory usage: 98.83MB