Cohomology of group number 933 of order 128

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t9  −  2·t8  +  2·t7  +  3·t5  +  2·t2  +  t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 12 minimal generators of maximal degree 8:

  1. a_1_1, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_0, an element of degree 1
  4. b_2_4, an element of degree 2
  5. b_2_5, an element of degree 2
  6. b_4_8, an element of degree 4
  7. b_5_9, an element of degree 5
  8. b_5_10, an element of degree 5
  9. b_5_11, an element of degree 5
  10. b_5_12, an element of degree 5
  11. b_5_13, an element of degree 5
  12. c_8_25, a Duflot regular element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 42 minimal relations of maximal degree 11:

  1. a_1_1·b_1_0
  2. a_1_2·b_1_0
  3. a_1_23 + a_1_13
  4. b_2_4·a_1_2 + a_1_1·a_1_22 + a_1_12·a_1_2 + a_1_13
  5. b_2_5·a_1_1 + a_1_1·a_1_22 + a_1_12·a_1_2 + a_1_13
  6. a_1_14
  7. a_1_13·a_1_2
  8. b_2_4·b_2_5 + a_1_12·a_1_22
  9. b_4_8·b_1_0
  10. b_1_0·b_5_9
  11. a_1_2·b_5_9 + a_1_1·b_5_10 + a_1_1·b_5_9 + b_4_8·a_1_22 + b_4_8·a_1_12
       + b_2_52·a_1_22
  12. b_1_0·b_5_10
  13. a_1_1·b_5_11 + a_1_1·b_5_9
  14. a_1_2·b_5_11 + a_1_2·b_5_9 + b_2_52·a_1_22
  15. b_1_0·b_5_12 + b_1_0·b_5_11 + b_2_42·a_1_12
  16. a_1_2·b_5_12 + b_4_8·a_1_22 + b_2_52·a_1_22
  17. a_1_1·b_5_13 + a_1_1·b_5_9 + b_4_8·a_1_1·a_1_2
  18. b_1_0·b_5_13 + b_2_52·a_1_22
  19. b_2_5·b_5_9 + b_2_4·b_5_10 + b_2_4·b_5_9 + b_2_5·b_4_8·a_1_2 + b_2_53·a_1_2
       + b_2_4·b_4_8·a_1_1 + b_4_8·a_1_13
  20. b_2_5·b_5_9 + b_2_5·b_4_8·a_1_2 + b_2_53·a_1_2 + a_1_1·a_1_2·b_5_10 + a_1_12·b_5_9
       + b_4_8·a_1_1·a_1_22 + b_4_8·a_1_13
  21. b_2_5·b_5_9 + b_2_5·b_4_8·a_1_2 + b_2_53·a_1_2 + a_1_22·b_5_10 + a_1_12·b_5_10
       + a_1_12·b_5_9 + b_4_8·a_1_1·a_1_22
  22. b_2_5·b_5_12 + b_2_5·b_5_11 + b_2_5·b_5_9 + b_2_5·b_4_8·a_1_2
  23. a_1_12·b_5_12 + b_4_8·a_1_12·a_1_2
  24. b_2_4·b_5_13 + b_2_4·b_5_9 + b_4_8·a_1_1·a_1_22 + b_4_8·a_1_12·a_1_2 + b_4_8·a_1_13
  25. b_2_5·b_5_9 + b_2_5·b_4_8·a_1_2 + b_2_53·a_1_2 + a_1_22·b_5_13 + a_1_12·b_5_10
       + b_4_8·a_1_12·a_1_2
  26. b_2_4·b_4_8·a_1_12 + a_1_13·b_5_9 + b_4_8·a_1_12·a_1_22
  27. b_2_5·b_4_8·a_1_22 + b_2_4·b_4_8·a_1_12 + a_1_13·b_5_10
  28. b_5_122 + b_5_112 + b_5_92 + b_2_43·b_4_8 + b_2_42·a_1_1·b_5_12
       + b_2_42·a_1_1·b_5_9 + b_4_82·a_1_22 + b_2_44·a_1_12
  29. b_5_11·b_5_12 + b_5_112 + b_5_10·b_5_12 + b_5_10·b_5_11 + b_5_9·b_5_10 + b_5_92
       + b_4_8·a_1_2·b_5_10 + b_4_8·a_1_1·b_5_10 + b_4_8·a_1_1·b_5_9 + b_2_42·a_1_1·b_5_9
       + b_4_82·a_1_22 + b_4_82·a_1_12 + b_2_54·a_1_22
  30. b_5_10·b_5_12 + b_5_10·b_5_11 + b_5_9·b_5_12 + b_5_9·b_5_10 + b_4_8·a_1_2·b_5_10
       + b_4_8·a_1_1·b_5_10 + b_4_8·a_1_1·b_5_9 + b_4_82·a_1_22 + b_4_82·a_1_12
       + b_2_54·a_1_22
  31. b_5_9·b_5_11 + b_5_92 + b_4_8·a_1_1·b_5_12 + b_4_82·a_1_1·a_1_2 + b_2_54·a_1_22
  32. b_5_132 + b_5_92 + b_2_53·b_4_8 + b_2_52·a_1_2·b_5_13 + b_2_52·a_1_2·b_5_10
       + b_4_82·a_1_22 + b_2_54·a_1_22
  33. b_5_132 + b_5_12·b_5_13 + b_5_10·b_5_12 + b_5_9·b_5_11 + b_2_53·b_4_8
       + b_4_8·a_1_2·b_5_10 + b_4_8·a_1_1·b_5_10 + b_4_8·a_1_1·b_5_9 + b_2_52·a_1_2·b_5_10
       + b_4_82·a_1_22 + b_4_82·a_1_12 + b_2_54·a_1_22
  34. b_5_11·b_5_13 + b_5_10·b_5_11 + b_5_9·b_5_10 + b_5_92 + b_4_8·a_1_1·b_5_10
       + b_4_8·a_1_1·b_5_9 + b_4_82·a_1_22 + b_4_82·a_1_12
  35. b_5_132 + b_5_10·b_5_11 + b_5_9·b_5_13 + b_5_9·b_5_11 + b_5_9·b_5_10 + b_5_92
       + b_2_53·b_4_8 + b_4_8·a_1_1·b_5_10 + b_4_8·a_1_1·b_5_9 + b_4_82·a_1_12
       + b_2_54·a_1_22
  36. b_5_10·b_5_11 + b_5_9·b_5_11 + b_5_9·b_5_10 + b_5_92 + b_4_8·a_1_2·b_5_13
       + b_4_8·a_1_1·b_5_10 + b_4_8·a_1_1·b_5_9 + b_2_52·a_1_2·b_5_10 + b_4_82·a_1_12
       + b_2_54·a_1_22
  37. b_5_92 + b_2_4·b_4_82 + b_4_8·a_1_1·b_5_9 + b_4_82·a_1_22 + b_4_82·a_1_1·a_1_2
       + b_2_54·a_1_22 + c_8_25·a_1_12
  38. b_5_112 + b_5_92 + b_1_05·b_5_11 + b_2_53·b_1_04 + b_2_4·b_1_03·b_5_11
       + b_2_54·a_1_22 + c_8_25·b_1_02
  39. b_5_9·b_5_10 + b_5_92 + b_4_8·a_1_2·b_5_10 + b_4_8·a_1_1·b_5_9 + b_2_52·a_1_2·b_5_10
       + b_4_82·a_1_12 + b_2_54·a_1_22 + c_8_25·a_1_1·a_1_2
  40. b_5_102 + b_5_92 + b_2_5·b_4_82 + b_4_8·a_1_2·b_5_10 + b_4_8·a_1_1·b_5_10
       + b_4_8·a_1_1·b_5_9 + b_4_82·a_1_22 + b_4_82·a_1_1·a_1_2 + c_8_25·a_1_22
  41. b_2_4·b_4_8·b_5_11 + b_2_4·b_4_8·b_5_9 + a_1_1·b_5_9·b_5_12 + b_4_8·a_1_12·b_5_10
       + b_4_8·a_1_12·b_5_9 + b_4_82·a_1_1·a_1_22 + b_4_82·a_1_13
  42. b_2_5·b_4_8·b_5_11 + a_1_2·b_5_10·b_5_13 + b_2_5·b_4_82·a_1_2
       + b_4_8·a_1_1·a_1_2·b_5_10 + b_4_8·a_1_12·b_5_9 + b_4_82·a_1_1·a_1_22
       + b_4_82·a_1_13 + c_8_25·a_1_1·a_1_22 + c_8_25·a_1_12·a_1_2

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 12.
  • However, the last relation was already found in degree 11 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_25, a Duflot regular element of degree 8
    2. b_1_04 + b_4_8 + b_2_52 + b_2_42, an element of degree 4
    3. b_2_5 + b_2_4, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. b_5_9 → 0, an element of degree 5
  8. b_5_10 → 0, an element of degree 5
  9. b_5_11 → 0, an element of degree 5
  10. b_5_12 → 0, an element of degree 5
  11. b_5_13 → 0, an element of degree 5
  12. c_8_25 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → c_1_1, an element of degree 1
  4. b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. b_5_9 → 0, an element of degree 5
  8. b_5_10 → 0, an element of degree 5
  9. b_5_11 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_12 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. b_5_13 → 0, an element of degree 5
  12. c_8_25 → c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → c_1_1, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_4_8 → 0, an element of degree 4
  7. b_5_9 → 0, an element of degree 5
  8. b_5_10 → 0, an element of degree 5
  9. b_5_11 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_5_12 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
       + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  11. b_5_13 → 0, an element of degree 5
  12. c_8_25 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
       + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. b_2_4 → c_1_12, an element of degree 2
  5. b_2_5 → 0, an element of degree 2
  6. b_4_8 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  7. b_5_9 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  8. b_5_10 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_5_11 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  10. b_5_12 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  11. b_5_13 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  12. c_8_25 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

  1. a_1_1 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_0 → 0, an element of degree 1
  4. b_2_4 → 0, an element of degree 2
  5. b_2_5 → c_1_12, an element of degree 2
  6. b_4_8 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  7. b_5_9 → 0, an element of degree 5
  8. b_5_10 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_5_11 → 0, an element of degree 5
  10. b_5_12 → 0, an element of degree 5
  11. b_5_13 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  12. c_8_25 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009