Cohomology of group number 621 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 4.
  • It is non-abelian.
  • It has p-Rank 5.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 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) · (t4  −  t3  +  t2  +  1)
    (t  +  1)2 · (t  −  1)5 · (t2  +  1)
  • 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 12 minimal generators of maximal degree 4:

  1. a_1_0, a nilpotent 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_6, an element of degree 2
  8. c_2_7, 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_4_22, an element of degree 4
  12. c_4_27, a Duflot regular element of degree 4

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Ring relations

There are 25 minimal relations of maximal degree 8:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_1_1·b_1_2 + a_1_0·b_1_2
  4. b_2_3·b_1_2
  5. b_2_3·a_1_0
  6. b_2_5·b_1_2 + b_2_4·b_1_2
  7. b_2_4·b_1_2 + b_2_5·a_1_0
  8. b_2_4·b_1_2 + b_2_6·a_1_0
  9. b_1_14 + b_2_32 + c_2_7·b_1_12
  10. b_1_2·b_3_13
  11. a_1_0·b_3_13
  12. b_1_1·b_3_13 + b_2_52 + b_2_4·b_2_6
  13. a_1_0·b_3_14
  14. b_1_1·b_3_14 + b_2_52 + b_2_4·b_2_6 + b_2_3·b_2_6 + b_2_3·b_2_5
  15. b_2_6·b_1_13 + b_2_5·b_1_13 + b_2_3·b_3_14 + b_2_3·b_3_13 + b_2_6·c_2_7·b_1_1
       + b_2_5·c_2_7·b_1_1
  16. b_4_22·b_1_2 + b_2_4·b_2_5·a_1_0
  17. b_4_22·a_1_0 + b_2_4·b_2_5·a_1_0
  18. b_4_22·b_1_1 + b_2_4·b_2_6·b_1_1 + b_2_3·b_3_13
  19. b_3_142 + b_3_132 + b_2_62·b_1_12 + b_2_52·b_1_12 + b_2_62·c_2_7
       + b_2_52·c_2_7
  20. b_3_132 + b_2_52·b_2_6 + b_2_53 + b_2_4·b_2_5·b_2_6 + b_2_4·b_2_52
       + c_4_27·b_1_12
  21. b_2_52·b_1_12 + b_2_4·b_2_6·b_1_12 + b_2_3·b_4_22 + b_2_3·b_2_4·b_2_6
       + b_2_52·c_2_7 + b_2_4·b_2_6·c_2_7
  22. b_3_13·b_3_14 + b_3_132 + b_2_6·b_4_22 + b_2_5·b_4_22 + b_2_4·b_2_62
       + b_2_4·b_2_5·b_2_6
  23. b_4_22·b_3_13 + b_2_4·b_2_6·b_3_14 + b_2_4·b_2_5·b_3_14 + b_2_4·b_2_5·b_3_13
       + b_2_3·b_2_6·b_3_13 + b_2_3·b_2_5·b_3_13 + b_2_3·c_4_27·b_1_1
  24. b_4_22·b_3_14 + b_2_52·b_2_6·b_1_1 + b_2_53·b_1_1 + b_2_4·b_2_6·b_3_13
       + b_2_4·b_2_62·b_1_1 + b_2_4·b_2_5·b_3_14 + b_2_4·b_2_5·b_3_13
       + b_2_4·b_2_5·b_2_6·b_1_1 + b_2_3·b_2_6·b_3_13 + b_2_3·b_2_5·b_3_13
       + b_2_6·c_2_7·b_3_13 + b_2_5·c_2_7·b_3_13 + b_2_3·c_4_27·b_1_1
  25. b_4_222 + b_2_4·b_2_62·b_1_12 + b_2_42·b_2_6·b_1_12 + b_2_42·b_2_62
       + b_2_3·b_2_6·b_4_22 + b_2_3·b_2_5·b_4_22 + b_2_3·b_2_4·b_4_22 + b_2_3·b_2_4·b_2_62
       + b_2_3·b_2_4·b_2_5·b_2_6 + b_2_3·b_2_42·b_2_6 + b_2_4·b_2_62·c_2_7
       + b_2_42·b_2_6·c_2_7 + b_2_32·c_4_27

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_7, a Duflot regular element of degree 2
    2. c_4_27, a Duflot regular element of degree 4
    3. b_1_24 + b_2_62 + b_2_4·b_2_6 + b_2_42 + b_2_32 + c_2_7·b_1_12, an element of degree 4
    4. b_2_62·b_1_22 + b_2_62·b_1_12 + b_2_4·b_2_6·b_1_12 + b_2_4·b_2_62
         + b_2_42·b_1_12 + b_2_42·b_2_6, 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].

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 2

  1. a_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_6 → 0, an element of degree 2
  8. c_2_7 → c_1_12, 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_4_22 → 0, an element of degree 4
  12. c_4_27 → c_1_04, an element of degree 4

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

  1. a_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_6 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  8. c_2_7 → c_1_12, an element of degree 2
  9. b_3_13 → 0, an element of degree 3
  10. b_3_14 → c_1_1·c_1_32 + c_1_1·c_1_2·c_1_3, an element of degree 3
  11. b_4_22 → 0, an element of degree 4
  12. c_4_27 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22 + c_1_04, an element of degree 4

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

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_4, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_2_3 → c_1_1·c_1_4, an element of degree 2
  5. b_2_4 → c_1_2·c_1_4 + c_1_22, an element of degree 2
  6. b_2_5 → c_1_3·c_1_4 + c_1_2·c_1_3 + c_1_0·c_1_4, an element of degree 2
  7. b_2_6 → c_1_3·c_1_4 + c_1_32, an element of degree 2
  8. c_2_7 → c_1_42 + c_1_12, an element of degree 2
  9. b_3_13 → c_1_32·c_1_4 + c_1_2·c_1_3·c_1_4 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_02·c_1_4, an element of degree 3
  10. b_3_14 → c_1_32·c_1_4 + c_1_2·c_1_3·c_1_4 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32
       + c_1_1·c_1_2·c_1_3 + c_1_0·c_1_1·c_1_4 + c_1_02·c_1_4, an element of degree 3
  11. b_4_22 → c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4 + c_1_22·c_1_3·c_1_4 + c_1_22·c_1_32
       + 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_02·c_1_1·c_1_4, an element of degree 4
  12. c_4_27 → 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_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_4 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22 + c_1_03·c_1_4 + c_1_04, 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