Magma V2.14-14 Fri Jan 02 2009 21:18:10 on havas-xps [Seed = 4108339778]
Type ? for help. Type -D to quit.
Construct a 2-generator anti-Hughes 7-group of class 14
which satisfies 6 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
B(2,7 q6 MO77 : 13) has order 7^141
The p-covering group G has order 7^208 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 "twB27c8q"
The quotient group H has order 7^159
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
Now factor out a complement for [b,a]^7 in the multiplier
Factoring out a complement of H.142 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^5
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^5
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^5
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^5
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73^5
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.143 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^4
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^4
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^4
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^4
Z, its centre, has order 7^6
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^72
generated by a and b; and [b,a]^7 = NextQ.72^4
The centre is generated by CurrentQ.72 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^72
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.144 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^6
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^6
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^6
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^6
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73^6
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.145 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^5
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^5
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^5
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^5
Z, its centre, has order 7^6
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^72
generated by a and b; and [b,a]^7 = NextQ.72^5
The centre is generated by CurrentQ.72 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^72
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.147 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^3
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^3
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^3
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73^3
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.148 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^2
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^2
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^2
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^2
Z, its centre, has order 7^6
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^72
generated by a and b; and [b,a]^7 = NextQ.72^2
The centre is generated by CurrentQ.72 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^72
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.150 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^4
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^4
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^4
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^4
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73^4
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.151 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77
Z, its centre, has order 7^7
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^71
generated by a and b; and [b,a]^7 = NextQ.71
The centre is generated by CurrentQ.71 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^71
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.152 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^4
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^4
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^4
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^4
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73^4
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.153 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^3
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^3
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^3
Z, its centre, has order 7^6
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^72
generated by a and b; and [b,a]^7 = NextQ.72^3
The centre is generated by CurrentQ.72 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^72
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.155 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^5
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^5
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^5
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^5
Z, its centre, has order 7^6
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^72
generated by a and b; and [b,a]^7 = NextQ.72^5
The centre is generated by CurrentQ.72 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^72
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.156 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.157 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^4
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^4
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^4
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^4
Z, its centre, has order 7^6
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^72
generated by a and b; and [b,a]^7 = NextQ.72^4
The centre is generated by CurrentQ.72 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^72
which is as far as we can reduce the group (by this method!)
Factoring out a complement of H.158 gives us a smaller anti-Hughes group
CurrentQ with order 7^142
CurrentQ is generated by a and b; and [b,a]^7 = CurrentQ.142^3
Z, its centre, has order 7^27
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^116
generated by a and b; and [b,a]^7 = NextQ.116^3
Z, its centre, has order 7^26
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^91
generated by a and b; and [b,a]^7 = NextQ.91^3
Z, its centre, has order 7^15
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^77
generated by a and b; and [b,a]^7 = NextQ.77^3
Z, its centre, has order 7^5
Now build a complement for [b,a]^7 in Z
Factor it out to get an anti-Hughes group of order 7^73
generated by a and b; and [b,a]^7 = NextQ.73^3
The centre is generated by CurrentQ.73 and has order 7^1
So this method reduces to an anti-Hughes group with order 7^73
which is as far as we can reduce the group (by this method!)
Total time: 10.781 seconds, Total memory usage: 31.95MB