GAP (crime) - Appendix B: Leisure and Recreation: Cohomology Rings of all Groups of Size 16

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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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