Magma V2.14-14 Fri Jan 02 2009 19:13:44 on havas-xps [Seed = 2713006644] Type ? for help. Type -D to quit. Construct a 3-generator 5-group anti-Hughes group of class 9 which satisfies 2 commutator defining relators and in which the normal closures of two generators have class 4 while the normal closure of the other generator, c, is abelian. Then repeatedly factor out complements to c^5 in the centre of the group, to obtain smaller counterexamples The class 8 quotient of the group has order 5^64 The p-covering group G has order 5^118 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 5^103 Now compute suitable 5th powers of elements outside G' Loading "twB35c5qmo441" The quotient group, H, has order 5^66 H is generated by a, b and c; and c^5 = H.66 H is anti-Hughes; the normal closures of a and b both have class 4 while 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' and c^5 generates gamma_9(H) Z, its centre, has order 5^8 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^59 generated by a, b and c; and c^5 = NextQ.59 Z, its centre, has order 5^7 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^53 generated by a, b and c; and c^5 = NextQ.53 Z, its centre, has order 5^5 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^49 generated by a, b and c; and c^5 = NextQ.49 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 5^48 generated by a, b and c; and c^5 = NextQ.48 The centre is generated by CurrentQ.48 and has order 5^1 So this method reduces to an anti-Hughes group with order 5^48 which is as far as we can reduce the group (by this method!) Total time: 0.984 seconds, Total memory usage: 25.10MB