Magma V2.14-14 Fri Jan 02 2009 10:53:16 on havas-xps [Seed = 1096117600] Type ? for help. Type -D to quit. Construct Khukhro's 3-generator 5-group counterexample to the Hughes conjecture, and then repeatedly factor out complements to c^5 in the centre of the group, to obtain smaller counterexamples B(3,5 : 8) has order 5^505 The p-covering group G has order 5^1380 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 Now compute suitable 5th powers of elements outside G' Loading "twB35c5" The quotient group, H, has order 5^917 H is generated by a, b and c; and c^5 = H.917 H is Khukhro's 3-generator anti-Hughes 5-group Z, its centre, has order 5^412 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^506 generated by a, b and c; and c^5 = NextQ.506 Z, its centre, has order 5^238 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^269 generated by a, b and c; and c^5 = NextQ.269 Z, its centre, has order 5^124 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^146 generated by a, b and c; and c^5 = NextQ.146 Z, its centre, has order 5^45 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^102 generated by a, b and c; and c^5 = NextQ.102 Z, its centre, has order 5^4 Now build a complement for c^5 in Z Factor it out to get an anti-Hughes group of order 5^99 generated by a, b and c; and c^5 = NextQ.99 The centre is generated by CurrentQ.99 and has order 5^1 So this method reduces to an anti-Hughes group with order 5^99 which is as far as we can reduce the group (by this method!) Total time: 128.312 seconds, Total memory usage: 632.31MB