Cohomology of group number 753 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its center has rank 2.
  • It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4, 4 and 5, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 5 and depth 3.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
     − 1
    (t  +  1)2 · (t  −  1)5
  • The a-invariants are -∞,-∞,-∞,-5,-5,-5. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 13 minimal generators of maximal degree 4:

  1. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_2_3, an element of degree 2
  5. b_2_4, an element of degree 2
  6. b_2_5, an element of degree 2
  7. b_2_7, an element of degree 2
  8. c_2_6, a Duflot regular element of degree 2
  9. b_3_13, an element of degree 3
  10. b_3_14, an element of degree 3
  11. b_3_15, an element of degree 3
  12. b_4_20, an element of degree 4
  13. c_4_29, a Duflot regular element of degree 4

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Ring relations

There are 34 minimal relations of maximal degree 8:

  1. b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. b_1_1·b_1_2
  4. b_2_3·b_1_2
  5. b_2_3·b_1_0
  6. b_2_5·b_1_2 + b_2_4·b_1_2
  7. b_2_5·b_1_0 + b_2_4·b_1_2
  8. b_2_7·b_1_0 + b_2_4·b_1_2
  9. b_2_32 + c_2_6·b_1_12
  10. b_1_2·b_3_13
  11. b_1_1·b_3_13 + b_2_3·b_2_5 + b_2_3·b_2_4
  12. b_1_2·b_3_14
  13. b_1_0·b_3_14
  14. b_1_1·b_3_14 + b_2_52 + b_2_4·b_2_7
  15. b_1_0·b_3_15
  16. b_1_1·b_3_15 + b_2_3·b_2_7 + b_2_3·b_2_5
  17. b_2_3·b_3_13 + b_2_5·c_2_6·b_1_1 + b_2_4·c_2_6·b_1_1
  18. b_2_3·b_3_15 + b_2_7·c_2_6·b_1_1 + b_2_5·c_2_6·b_1_1
  19. b_2_7·b_3_13 + b_2_5·b_3_15 + b_2_3·b_3_14
  20. b_2_5·b_3_13 + b_2_4·b_3_15 + b_2_3·b_3_14
  21. b_4_20·b_1_2
  22. b_4_20·b_1_0
  23. b_4_20·b_1_1 + b_2_3·b_3_14
  24. b_3_13·b_3_15 + b_2_5·b_2_7·c_2_6 + b_2_52·c_2_6 + b_2_4·b_2_7·c_2_6
       + b_2_4·b_2_5·c_2_6
  25. b_3_152 + b_2_7·b_1_2·b_3_15 + c_4_29·b_1_22 + b_2_72·c_2_6 + b_2_52·c_2_6
  26. b_3_132 + b_2_4·b_1_0·b_3_13 + c_4_29·b_1_02 + b_2_52·c_2_6 + b_2_42·c_2_6
  27. b_3_142 + b_2_52·b_1_12 + b_2_52·b_2_7 + b_2_53 + b_2_4·b_2_5·b_1_12
       + b_2_4·b_2_5·b_2_7 + b_2_4·b_2_52 + c_4_29·b_1_12
  28. b_2_3·b_4_20 + b_2_52·c_2_6 + b_2_4·b_2_7·c_2_6
  29. b_3_13·b_3_14 + b_2_5·b_4_20 + b_2_4·b_4_20
  30. b_3_14·b_3_15 + b_2_7·b_4_20 + b_2_5·b_4_20
  31. b_4_20·b_3_13 + b_2_5·c_2_6·b_3_14 + b_2_4·c_2_6·b_3_14
  32. b_4_20·b_3_15 + b_2_7·c_2_6·b_3_14 + b_2_5·c_2_6·b_3_14
  33. b_4_20·b_3_14 + b_2_4·b_2_7·b_3_15 + b_2_4·b_2_5·b_3_15 + b_2_3·b_2_7·b_3_14
       + b_2_3·b_2_5·b_3_14 + b_2_3·b_2_52·b_1_1 + b_2_3·b_2_4·b_2_5·b_1_1
       + b_2_3·c_4_29·b_1_1
  34. b_4_202 + b_2_52·c_2_6·b_1_12 + b_2_52·b_2_7·c_2_6 + b_2_53·c_2_6
       + b_2_4·b_2_5·c_2_6·b_1_12 + b_2_4·b_2_5·b_2_7·c_2_6 + b_2_4·b_2_52·c_2_6
       + c_2_6·c_4_29·b_1_12

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 13.
  • However, the last relation was already found in degree 8 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_6, a Duflot regular element of degree 2
    2. c_4_29, a Duflot regular element of degree 4
    3. b_1_24 + b_1_14 + b_1_04 + b_2_72 + b_2_4·b_2_7 + b_2_42, an element of degree 4
    4. b_2_72·b_1_22 + b_2_72·b_1_12 + b_2_4·b_2_7·b_1_12 + b_2_4·b_2_72
         + b_2_42·b_1_12 + b_2_42·b_1_02 + b_2_42·b_2_7, an element of degree 6
    5. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 11, 13].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_7 → 0, an element of degree 2
  8. c_2_6 → c_1_02, an element of degree 2
  9. b_3_13 → 0, an element of degree 3
  10. b_3_14 → 0, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. b_4_20 → 0, an element of degree 4
  13. c_4_29 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → c_1_2, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_7 → 0, an element of degree 2
  8. c_2_6 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  9. b_3_13 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_32 + c_1_0·c_1_2·c_1_3, an element of degree 3
  10. b_3_14 → 0, an element of degree 3
  11. b_3_15 → 0, an element of degree 3
  12. b_4_20 → 0, an element of degree 4
  13. c_4_29 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_2_3 → 0, an element of degree 2
  5. b_2_4 → 0, an element of degree 2
  6. b_2_5 → 0, an element of degree 2
  7. b_2_7 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  8. c_2_6 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  9. b_3_13 → 0, an element of degree 3
  10. b_3_14 → 0, an element of degree 3
  11. b_3_15 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_32 + c_1_0·c_1_2·c_1_3, an element of degree 3
  12. b_4_20 → 0, an element of degree 4
  13. c_4_29 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
       + c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 5

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → c_1_0·c_1_2, an element of degree 2
  5. b_2_4 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  6. b_2_5 → c_1_3·c_1_4 + c_1_1·c_1_2, an element of degree 2
  7. b_2_7 → c_1_42 + c_1_2·c_1_4, an element of degree 2
  8. c_2_6 → c_1_02, an element of degree 2
  9. b_3_13 → c_1_0·c_1_3·c_1_4 + c_1_0·c_1_32 + c_1_0·c_1_2·c_1_3 + c_1_0·c_1_1·c_1_2, an element of degree 3
  10. b_3_14 → c_1_3·c_1_42 + c_1_32·c_1_4 + c_1_2·c_1_3·c_1_4 + c_1_12·c_1_2, an element of degree 3
  11. b_3_15 → c_1_0·c_1_42 + c_1_0·c_1_3·c_1_4 + c_1_0·c_1_2·c_1_4 + c_1_0·c_1_1·c_1_2, an element of degree 3
  12. b_4_20 → c_1_0·c_1_3·c_1_42 + c_1_0·c_1_32·c_1_4 + c_1_0·c_1_2·c_1_3·c_1_4
       + c_1_0·c_1_12·c_1_2, an element of degree 4
  13. c_4_29 → c_1_32·c_1_42 + c_1_33·c_1_4 + c_1_2·c_1_32·c_1_4 + c_1_1·c_1_3·c_1_42
       + c_1_1·c_1_32·c_1_4 + c_1_1·c_1_2·c_1_3·c_1_4 + c_1_1·c_1_2·c_1_32
       + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_42 + c_1_12·c_1_3·c_1_4 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_4 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_14, an element of degree 4

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