Magma V2.14-14 Fri Jan 02 2009 18:48:35 on havas-xps [Seed = 2643001379] 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 closure of one 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 5^228 The p-covering group G has order 5^567 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 The group now has order 5^141 Now compute suitable 5th powers of elements outside G' Loading "twB35c5q" The quotient group, H, has order 5^87 H is generated by a, b and c; and c^5 = H.87 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 5^17 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^71 generated by a, b and c; and c^5 = NextQ.71 Z, its centre, has order 5^13 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 5^9 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^51 generated by a, b and c; and c^5 = NextQ.51 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 5^50 generated by a, b and c; and c^5 = NextQ.50 The centre is generated by CurrentQ.50 and has order 5^1 So this method reduces to an anti-Hughes group with order 5^50 which is as far as we can reduce the group (by this method!) Total time: 59.343 seconds, Total memory usage: 45.91MB