Magma V2.14-14 Sun Jan 04 2009 10:53:57 on havas-xps [Seed = 1918579593] Type ? for help. Type -D to quit. Construct the largest class 9 quotient of Wall's 3-generator anti-Hughes 5-group. Then repeatedly factor out complements to c^5 in the centre of the group, to obtain smaller counterexamples The p-covering group G has order 5^283 and class 9 G is generated by a, b and c; and c has order 25 [b,a] has order 5 ; [c,a] has order 5 ; [c,b] has order 5 gamma_2(G) is the normal closure of < [b,a], [c,a], [c,b] >, and gamma_2(G) has class at most 4, so gamma_2(G) has exponent 5 Also factor out the derived group of the normal closure of c (This became non-trivial in the p-cover computation) The group now has order 5^227 Now compute suitable 5th powers of elements outside G' Loading "twB35c5mo441" The quotient group H has order 5^151 H is generated by a, b and c; and c^5 = H.151 H is anti-Hughes; the normal closure of c is abelian and this implies that gamma_3(G) has class 3 so every element cg, g in gamma_3, has order 25 We also have: c[b,a] has order 25 ; c[c,a] has order 25 ; c[c,b] has order 25 So the Hughes subgroup is G' Now factor out a complement for c^5 in the multiplier Factoring out a complement of H.151 gives us a smaller anti-Hughes group CurrentQ with order 5^118 CurrentQ is generated by a, b and c; and c^5 = CurrentQ.118 Z, its centre, has order 5^29 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^90 generated by a, b and c; and c^5 = NextQ.90 Z, its centre, has order 5^24 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^67 generated by a, b and c; and c^5 = NextQ.67 Z, its centre, has order 5^11 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^57 generated by a, b and c; and c^5 = NextQ.57 Z, its centre, has order 5^2 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^56 generated by a, b and c; and c^5 = NextQ.56 The centre is generated by CurrentQ.56 and has order 5^1 So this method reduces to an anti-Hughes group with order 5^56 which is as far as we can reduce the group (by this method!) Total time: 8.437 seconds, Total memory usage: 28.00MB