Magma V2.14-14 Fri Jan 02 2009 11:12:14 on havas-xps [Seed = 2409344806]
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Construct a 3-generator anti-Hughes 5-group of class 9 in
which the normal closures of two generators have class 4 while
the normal closure of the other 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 5^109
The p-covering group G has order 5^221 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 5^171
Now compute suitable 5th powers of elements outside G'
Loading "twB35c5mo441"
The quotient group, H, has order 5^123
H is generated by a, b and c; and c^5 = H.123
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 5^14
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^110
generated by a, b and c; and c^5 = NextQ.110
Z, its centre, has order 5^23
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^88
generated by a, b and c; and c^5 = NextQ.88
Z, its centre, has order 5^22
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 5^12
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
Z, its centre, has order 5^3
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^54
generated by a, b and c; and c^5 = NextQ.54
The centre is generated by CurrentQ.54 and has order 5^1
So this method reduces to an anti-Hughes group with order 5^54
which is as far as we can reduce the group (by this method!)
Total time: 2.968 seconds, Total memory usage: 25.13MB