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3. Leisure and Recreation: Cohomology Rings of all Groups of Size 16

### 3. 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, and runs `ProjectiveResolution`, `CohomologyGenerators` and `CohomologyRelators` for each group, and prints the results as well as the timings for each operation to a file. The output below was computed on a 3.06 GHz Intel processor with 3.71 GB of RAM. The projective resolutions are calculated initially to degree 10 and the generators and relators to degree 6, due to the fact that I already knew all the generators and relators to be of degree less than 6, see http://www.math.uga.edu/~lvalero/cohointro.html. See also 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 ]
    Time:  0:00:05.864
    Generators in degrees: [ 1, 2 ]
    Time:  0:00:00.086
    Relators: [ [ z, y ], [ z^2 ] ]
    Time:  0:00:00.245

    SmallGroup(16,2)
    Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
    Time:  0:00:00.931
    Generators in degrees: [ 1, 1, 2, 2 ]
    Time:  0:00:02.874
    Relators: [ [ z, y, x, w ], [ z^2, y^2 ] ]
    Time:  0:00:12.227

    SmallGroup(16,3)
    Betti Numbers: [ 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36 ]
    Time:  0:00:05.292
    Generators in degrees: [ 1, 1, 2, 2, 2 ]
    Time:  0:00:21.770
    Relators: [ [ z, y, x, w, v ], [ z^2, z*y, z*x, y^2*v+x^2 ] ]
    Time:  0:01:26.166

    SmallGroup(16,4)
    Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
    Time:  0:00:01.047
    Generators in degrees: [ 1, 1, 2, 2 ]
    Time:  0:00:03.253
    Relators: [ [ z, y, x, w ], [ z^2, z*y+y^2, y^3 ] ]
    Time:  0:00:14.294

    SmallGroup(16,5)
    Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
    Time:  0:00:01.065
    Generators in degrees: [ 1, 1, 2 ]
    Time:  0:00:02.493
    Relators: [ [ z, y, x ], [ z^2 ] ]
    Time:  0:00:13.573

    SmallGroup(16,6)
    Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ]
    Time:  0:00:00.446
    Generators in degrees: [ 1, 1, 3, 4 ]
    Time:  0:00:01.566
    Relators: [ [ z, y, x, w ], [ z^2, z*y^2, z*x, x^2 ] ]
    Time:  0:00:04.132

    SmallGroup(16,7)
    Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ]
    Time:  0:00:01.076
    Generators in degrees: [ 1, 1, 2 ]
    Time:  0:00:02.495
    Relators: [ [ z, y, x ], [ z*y ] ]
    Time:  0:00:13.862

    SmallGroup(16,8)
    Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ]
    Time:  0:00:00.465
    Generators in degrees: [ 1, 1, 3, 4 ]
    Time:  0:00:01.570
    Relators: [ [ z, y, x, w ], [ z*y, z^3, z*x, y^2*w+x^2 ] ]
    Time:  0:00:04.350

    SmallGroup(16,9)
    Betti Numbers: [ 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2 ]
    Time:  0:00:00.140
    Generators in degrees: [ 1, 1, 4 ]
    Time:  0:00:00.255
    Relators: [ [ z, y, x ], [ z*y, z^3+y^3, y^4 ] ]
    Time:  0:00:00.718

    SmallGroup(16,10)
    Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ]
    Time:  0:00:20.139
    Generators in degrees: [ 1, 1, 1, 2 ]
    Time:  0:01:04.158
    Relators: [ [ z, y, x, w ], [ z^2 ] ]
    Time:  0:06:27.688

    SmallGroup(16,11)
    Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ]
    Time:  0:00:20.428
    Generators in degrees: [ 1, 1, 1, 2 ]
    Time:  0:01:04.678
    Relators: [ [ z, y, x, w ], [ z*y ] ]
    Time:  0:06:33.808

    SmallGroup(16,12)
    Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ]
    Time:  0:00:02.438
    Generators in degrees: [ 1, 1, 1, 4 ]
    Time:  0:00:08.927
    Relators: [ [ z, y, x, w ], [ z^2+z*y+y^2, y^3 ] ]
    Time:  0:00:44.464

    SmallGroup(16,13)
    Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ]
    Time:  0:00:02.389
    Generators in degrees: [ 1, 1, 1, 4 ]
    Time:  0:00:09.247
    Relators: [ [ z, y, x, w ], [ z*y+x^2, z*x^2+y*x^2, y^2*x^2+x^4 ] ]
    Time:  0:00:44.323

    SmallGroup(16,14)
    Betti Numbers: [ 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286 ]
    Time:  0:07:00.973
    Generators in degrees: [ 1, 1, 1, 1 ]
    Time:  0:15:40.874
    Relators: [ [ z, y, x, w ], [  ] ]
    Time:  1:54:28.052

    Total time:  2:38:14.841

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