Magma V2.14-14 Sun Jan 04 2009 10:53:57 on havas-xps [Seed = 1918579593]
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Construct the largest class 9 quotient of Wall's 3-generator
anti-Hughes 5-group.
Then repeatedly factor out complements to c^5 in the centre
of the group, to obtain smaller counterexamples
The p-covering group G has order 5^283 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 5^227
Now compute suitable 5th powers of elements outside G'
Loading "twB35c5mo441"
The quotient group H has order 5^151
H is generated by a, b and c; and c^5 = H.151
H is anti-Hughes; 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'
Now factor out a complement for c^5 in the multiplier
Factoring out a complement of H.151 gives us a smaller anti-Hughes group
CurrentQ with order 5^118
CurrentQ is generated by a, b and c; and c^5 = CurrentQ.118
Z, its centre, has order 5^29
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^90
generated by a, b and c; and c^5 = NextQ.90
Z, its centre, has order 5^24
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^67
generated by a, b and c; and c^5 = NextQ.67
Z, its centre, has order 5^11
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^57
generated by a, b and c; and c^5 = NextQ.57
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 5^56
generated by a, b and c; and c^5 = NextQ.56
The centre is generated by CurrentQ.56 and has order 5^1
So this method reduces to an anti-Hughes group with order 5^56
which is as far as we can reduce the group (by this method!)
Total time: 8.437 seconds, Total memory usage: 28.00MB