Magma V2.14-14 Fri Jan 02 2009 18:48:35 on havas-xps [Seed = 2643001379]
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Construct a 3-generator 5-group anti-Hughes group of class 9
which satisfies 2 commutator defining relators and in which
the normal closure of one 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 5^228
The p-covering group G has order 5^567 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
The group now has order 5^141
Now compute suitable 5th powers of elements outside G'
Loading "twB35c5q"
The quotient group, H, has order 5^87
H is generated by a, b and c; and c^5 = H.87
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 5^17
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^71
generated by a, b and c; and c^5 = NextQ.71
Z, its centre, has order 5^13
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^59
generated by a, b and c; and c^5 = NextQ.59
Z, its centre, has order 5^9
Now build a complement for c^5 in Z
Factor it out to get an anti-Hughes group of order 5^51
generated by a, b and c; and c^5 = NextQ.51
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 5^50
generated by a, b and c; and c^5 = NextQ.50
The centre is generated by CurrentQ.50 and has order 5^1
So this method reduces to an anti-Hughes group with order 5^50
which is as far as we can reduce the group (by this method!)
Total time: 59.343 seconds, Total memory usage: 45.91MB