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B Leisure and Recreation: Cohomology Rings of all Groups of Size 16

B Leisure and Recreation: Cohomology Rings of all Groups of Size 16

Below is the output of the test file tst/batch.g. The file runs through all groups of size n, which is initially set to 16, calls ProjectiveResolution, CohomologyGenerators, and CohomologyRelators for each group, and prints the results, as well as the runtimes for each operation, to a file like the one shown below. The runtimes in this example have been deleted, having been presented in Appendix A. The example below was computed on a 2.4 GHz AMD64 processor with 12 GB of RAM. See the file tst/README for suggestions on dealing with other users when running long-running batch processes.


SmallGroup(16,1)
Betti Numbers: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
Generators in degrees: [ 1, 2 ]
Relators: [ [ z, y ], [ z^2 ] ]

SmallGroup(16,2)
Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
Generators in degrees: [ 1, 1, 2, 2 ]
Relators: [ [ z, y, x, w ], [ z^2, y^2 ] ]

SmallGroup(16,3)
Betti Numbers: [ 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36 ]
Generators in degrees: [ 1, 1, 2, 2, 2 ]
Relators: [ [ z, y, x, w, v ], [ z^2, z*y, z*x, y^2*v+x^2 ] ]

SmallGroup(16,4)
Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
Generators in degrees: [ 1, 1, 2, 2 ]
Relators: [ [ z, y, x, w ], [ z^2, z*y+y^2, y^3 ] ]

SmallGroup(16,5)
Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
Generators in degrees: [ 1, 1, 2 ]
Relators: [ [ z, y, x ], [ z^2 ] ]

SmallGroup(16,6)
Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ]
Generators in degrees: [ 1, 1, 3, 4 ]
Relators: [ [ z, y, x, w ], [ z^2, z*y^2, z*x, x^2 ] ]

SmallGroup(16,7)
Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
Generators in degrees: [ 1, 1, 2 ]
Relators: [ [ z, y, x ], [ z*y ] ]

SmallGroup(16,8)
Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ]
Generators in degrees: [ 1, 1, 3, 4 ]
Relators: [ [ z, y, x, w ], [ z*y, z^3, z*x, y^2*w+x^2 ] ]

SmallGroup(16,9)
Betti Numbers: [ 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2 ]
Generators in degrees: [ 1, 1, 4 ]
Relators: [ [ z, y, x ], [ z*y, z^3+y^3, y^4 ] ]

SmallGroup(16,10)
Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ]
Generators in degrees: [ 1, 1, 1, 2 ]
Relators: [ [ z, y, x, w ], [ z^2 ] ]

SmallGroup(16,11)
Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ]
Generators in degrees: [ 1, 1, 1, 2 ]
Relators: [ [ z, y, x, w ], [ z*y ] ]

SmallGroup(16,12)
Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ]
Generators in degrees: [ 1, 1, 1, 4 ]
Relators: [ [ z, y, x, w ], [ z^2+z*y+y^2, y^3 ] ]

SmallGroup(16,13)
Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ]
Generators in degrees: [ 1, 1, 1, 4 ]
Relators: [ [ z, y, x, w ], [ z*y+x^2, z*x^2+y*x^2, y^2*x^2+x^4 ] ]

SmallGroup(16,14)
Betti Numbers: [ 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286 ]
Generators in degrees: [ 1, 1, 1, 1 ]
Relators: [ [ z, y, x, w ], [  ] ]

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