Magma V2.14-14 Fri Jan 02 2009 21:36:46 on havas-xps [Seed = 1769856098] Type ? for help. Type -D to quit. Construct a 2-generator anti-Hughes 7-group of class 14 in which the normal closures of the generators both have class 7. Then repeatedly factor out complements to [b,a]^7 in the centre of the group, to obtain smaller counterexamples B(2,7 MO77 : 13) has order 7^488 The p-covering group G has order 7^680 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 7^597 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.489 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^3 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.490 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.491 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^4 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.492 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.493 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.494 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.496 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.497 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.498 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.499 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.500 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.501 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.502 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.503 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.504 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^2 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.505 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.506 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^2 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.507 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.508 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.509 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^2 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.510 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.511 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^120 generated by a and b; and [b,a]^7 = NextQ.120^6 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.512 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.514 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.516 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.517 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.518 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.519 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.520 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.521 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.522 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.523 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119^4 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.524 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.525 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.526 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.527 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.528 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.529 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^2 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.530 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119^3 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.531 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.532 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119^4 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.533 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.534 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^120 generated by a and b; and [b,a]^7 = NextQ.120^6 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.536 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^5 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.537 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^2 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.539 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.541 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.542 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^120 generated by a and b; and [b,a]^7 = NextQ.120^6 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.543 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^3 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.544 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.545 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.546 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119^3 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.547 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.549 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^3 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.551 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^5 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.553 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119^2 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.555 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.556 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^2 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.557 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.558 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^2 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.559 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.560 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.562 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.563 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.564 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.566 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.567 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.569 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.570 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.571 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.572 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.574 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^2 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.575 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^3 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.576 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^2 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.579 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.581 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^6 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^120 generated by a and b; and [b,a]^7 = NextQ.120^6 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.582 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.583 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^2 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.586 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.587 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.588 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168 Z, its centre, has order 7^44 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 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.590 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^2 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^2 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^2 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^2 Z, its centre, has order 7^44 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.125^2 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^2 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.591 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.592 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^4 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.593 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^120 generated by a and b; and [b,a]^7 = NextQ.120^5 The centre is generated by CurrentQ.120 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^120 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.594 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^3 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^3 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^3 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^3 Z, its centre, has order 7^44 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.125^3 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^3 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.595 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^5 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^5 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^5 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^5 Z, its centre, has order 7^44 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.125^5 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 7^121 generated by a and b; and [b,a]^7 = NextQ.121^5 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.596 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^6 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^6 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^6 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^6 Z, its centre, has order 7^44 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 7^5 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^121 generated by a and b; and [b,a]^7 = NextQ.121^6 The centre is generated by CurrentQ.121 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^121 which is as far as we can reduce the group (by this method!) Factoring out a complement of H.597 gives us a smaller anti-Hughes group CurrentQ with order 7^489 CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.489^4 Z, its centre, has order 7^133 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^357 generated by a and b; and [b,a]^7 = NextQ.357^4 Z, its centre, has order 7^114 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^244 generated by a and b; and [b,a]^7 = NextQ.244^4 Z, its centre, has order 7^77 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^168 generated by a and b; and [b,a]^7 = NextQ.168^4 Z, its centre, has order 7^44 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.125^4 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 7^119 generated by a and b; and [b,a]^7 = NextQ.119^4 The centre is generated by CurrentQ.119 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^119 which is as far as we can reduce the group (by this method!) Total time: 709.625 seconds, Total memory usage: 150.43MB