Magma V2.14-14 Sun Jan 04 2009 10:17:28 on havas-xps [Seed = 3788023972] Type ? for help. Type -D to quit. Construct a 2-generator anti-Hughes 7-group of class 14 which satisfies 11 commutator defining relators and 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 The class 13 quotient of the group has order 7^96 The p-covering group G has order 7^143 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 "tw7.m" The quotient group H has order 7^97 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 [b,a]^7 generates gamma_14(H) The centre, Z, of the current group has order 7^11 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^87 generated by a and b; and [b,a]^7 = NextQ.87^4 The centre, Z, of the current group has order 7^10 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^78 generated by a and b; and [b,a]^7 = NextQ.78^4 The centre, Z, of the current group has order 7^10 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^69 generated by a and b; and [b,a]^7 = NextQ.69^4 The centre, Z, of the current group has order 7^4 Now build a complement for [b,a]^7 in Z Factor it out to get an anti-Hughes group of order 7^66 generated by a and b; and [b,a]^7 = NextQ.66^4 The centre is generated by CurrentQ.66 and has order 7^1 So this method reduces to an anti-Hughes group with order 7^66 which is as far as we can reduce the group (by this method!) Total time: 5.531 seconds, Total memory usage: 29.57MB