Magma V2.14-14 Fri Jan 02 2009 20:22:59 on havas-xps [Seed = 3453579159]
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Construct Khukhro's 2-generator 7-group counterexample to the Hughes
conjecture, and then repeatedly factor out complements to [b,a]^7 in
the centre of the group, so as to obtain smaller counterexamples
B(2,7 : 13) has order 7^668
The p-covering group G has order 7^1258 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 "twB27c8"
The quotient group H has order 7^1075
H is generated by a and b; and [b,a] has order 49
H is Khukhro's 2-generator anti-Hughes 7-group
Now factor out a complement for [b,a]^7 in the multiplier
Factoring out a complement of H.846 gives us a smaller anti-Hughes group
CurrentQ with order 7^669
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.669^6
Z, its centre, has order 7^259
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^411
generated by a and b; and [b,a]^7 = NextQ.411^6
Z, its centre, has order 7^156
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^256
generated by a and b; and [b,a]^7 = NextQ.256^6
Z, its centre, has order 7^87
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^170
generated by a and b; and [b,a]^7 = NextQ.170^6
Z, its centre, has order 7^46
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^125
generated by a and b; and [b,a]^7 = NextQ.125^6
Z, its centre, has order 7^9
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^117
generated by a and b; and [b,a]^7 = NextQ.117^6
The centre is generated by CurrentQ.117 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^117
which is as far as we can reduce the group (by this method!)
Total time: 151.468 seconds, Total memory usage: 734.27MB