Cohomology of group number 1045 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 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 3.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    (2) · (t3  +  1/2·t  +  1/2)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-4,-4. 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 5:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_1_2, a nilpotent element of degree 1
  4. b_1_3, an element of degree 1
  5. c_2_7, a Duflot regular element of degree 2
  6. a_3_9, a nilpotent element of degree 3
  7. a_3_11, a nilpotent element of degree 3
  8. a_3_10, a nilpotent element of degree 3
  9. b_3_13, an element of degree 3
  10. b_4_20, an element of degree 4
  11. c_4_22, a Duflot regular element of degree 4
  12. c_4_23, a Duflot regular element of degree 4
  13. b_5_37, an element of degree 5

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

Ring relations

There are 38 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_12 + a_1_0·a_1_2
  3. a_1_0·b_1_3 + a_1_22 + a_1_1·a_1_2 + a_1_0·a_1_1
  4. a_1_0·a_1_1·a_1_2
  5. a_1_22·b_1_3 + a_1_1·a_1_2·b_1_3
  6. a_1_0·a_3_9
  7. a_1_2·a_3_9 + a_1_1·a_3_11 + a_1_1·a_3_9
  8. a_1_1·a_3_9 + a_1_0·a_3_11
  9. a_1_2·a_3_11 + a_1_2·a_3_9 + a_1_1·a_3_10
  10. a_1_2·a_3_9 + a_1_0·a_3_10
  11. b_1_3·a_3_9 + a_1_2·a_3_10 + a_1_2·a_3_11 + a_1_2·a_3_9 + a_1_1·a_3_9
  12. a_1_0·b_3_13
  13. a_1_2·b_1_3·a_3_10 + a_1_1·b_1_3·a_3_10 + a_1_1·a_1_2·b_3_13 + a_1_0·a_1_1·a_3_10
  14. b_1_32·a_3_11 + a_1_1·b_1_3·b_3_13 + b_4_20·a_1_1 + a_1_1·b_1_3·a_3_10 + a_1_22·a_3_10
       + a_1_0·a_1_1·a_3_10 + c_2_7·a_1_1·b_1_32 + c_2_7·a_1_1·a_1_22
  15. b_4_20·a_1_0 + a_1_22·a_3_10 + a_1_1·a_1_2·a_3_10 + a_1_0·a_1_1·a_3_10
       + c_2_7·a_1_1·a_1_22
  16. b_1_32·a_3_10 + a_1_2·b_1_3·b_3_13 + a_1_1·b_1_3·b_3_13 + b_4_20·a_1_2
       + a_1_1·a_1_2·a_3_10 + a_1_0·a_1_1·a_3_10 + c_2_7·a_1_2·b_1_32 + c_2_7·a_1_23
  17. a_3_92
  18. a_3_112 + a_3_9·a_3_10
  19. a_3_9·b_3_13 + c_2_7·a_1_1·a_1_2·b_1_32
  20. b_3_132 + c_2_7·b_1_34
  21. a_3_9·a_3_11 + c_4_22·a_1_0·a_1_1
  22. a_3_11·a_3_10 + a_3_112 + a_1_1·a_1_22·a_3_10 + c_4_22·a_1_1·a_1_2
       + c_2_7·a_1_1·a_1_2·b_1_32
  23. a_3_112 + c_4_22·a_1_0·a_1_2
  24. a_3_102 + c_4_22·a_1_22 + c_2_7·a_1_1·a_1_2·b_1_32
  25. a_3_11·b_3_13 + a_1_1·b_5_37 + a_3_112 + a_1_1·a_1_2·b_1_3·b_3_13 + c_4_23·a_1_1·b_1_3
       + c_2_7·a_1_1·b_1_33 + c_4_23·a_1_1·a_1_2 + c_4_23·a_1_0·a_1_2
       + c_2_7·a_1_1·a_1_2·b_1_32 + c_2_7·a_1_0·a_3_11 + c_2_72·a_1_1·b_1_3
       + c_2_72·a_1_0·a_1_2
  26. a_1_0·b_5_37 + a_3_9·a_3_11 + c_4_23·a_1_22 + c_4_23·a_1_1·a_1_2 + c_4_23·a_1_0·a_1_2
       + c_2_72·a_1_22 + c_2_72·a_1_1·a_1_2
  27. a_3_10·b_3_13 + a_1_2·b_5_37 + a_3_11·a_3_10 + a_3_112 + a_1_1·a_1_2·b_1_3·b_3_13
       + b_4_20·a_1_1·a_1_2 + c_4_23·a_1_2·b_1_3 + c_2_7·a_1_2·b_1_33 + c_2_7·a_1_1·b_1_33
       + c_4_23·a_1_22 + c_4_23·a_1_1·a_1_2 + c_2_7·a_1_0·a_3_10 + c_2_72·a_1_2·b_1_3
       + c_2_72·a_1_1·a_1_2
  28. b_1_32·b_5_37 + b_4_20·b_3_13 + a_1_2·b_1_33·b_3_13 + b_4_20·a_3_10
       + b_4_20·a_1_1·b_1_32 + b_4_20·a_1_1·a_1_2·b_1_3 + c_4_23·b_1_33
       + c_2_7·b_1_32·b_3_13 + c_4_23·a_1_2·b_1_32 + c_4_23·a_1_1·b_1_32
       + c_4_22·a_1_2·b_1_32 + c_4_22·a_1_1·b_1_32 + c_2_7·a_1_2·b_1_3·b_3_13
       + c_2_7·a_1_1·b_1_3·b_3_13 + c_2_7·a_1_1·b_1_34 + c_2_7·b_4_20·a_1_2
       + c_4_22·a_1_1·a_1_22 + c_2_7·a_1_22·a_3_10 + c_2_72·b_1_33
       + c_2_72·a_1_2·b_1_32 + c_2_72·a_1_1·b_1_32 + c_2_72·a_1_23
  29. a_1_1·b_1_3·b_5_37 + b_4_20·a_3_11 + a_1_1·a_1_2·b_1_32·b_3_13
       + c_4_23·a_1_1·b_1_32 + c_4_22·a_1_1·b_1_32 + c_2_7·a_1_1·b_1_3·b_3_13
       + c_2_7·b_4_20·a_1_1 + c_4_23·a_1_1·a_1_2·b_1_3 + c_4_22·a_1_1·a_1_2·b_1_3
       + c_2_7·a_1_1·b_1_3·a_3_10 + c_4_23·a_1_23 + c_4_23·a_1_1·a_1_22 + c_4_22·a_1_23
       + c_4_22·a_1_1·a_1_22 + c_2_7·a_1_22·a_3_10 + c_2_72·a_1_23
  30. a_1_2·b_1_3·b_5_37 + b_4_20·a_3_10 + b_4_20·a_3_11 + b_4_20·a_3_9
       + a_1_1·a_1_2·b_1_32·b_3_13 + b_4_20·a_1_1·a_1_2·b_1_3 + c_4_23·a_1_2·b_1_32
       + c_4_22·a_1_2·b_1_32 + c_4_22·a_1_1·b_1_32 + c_2_7·a_1_2·b_1_3·b_3_13
       + c_2_7·b_4_20·a_1_2 + c_2_7·b_4_20·a_1_1 + c_2_7·a_1_1·b_1_3·a_3_10
       + c_2_7·a_1_1·a_1_2·b_3_13 + c_2_7·a_1_1·a_1_2·b_1_33 + c_4_22·a_1_23
       + c_2_7·a_1_22·a_3_10 + c_2_7·a_1_0·a_1_1·a_3_10 + c_2_72·a_1_1·b_1_32
       + c_2_72·a_1_1·a_1_2·b_1_3 + c_2_72·a_1_23 + c_2_72·a_1_1·a_1_22
  31. b_4_20·a_3_9 + a_1_1·a_1_2·b_5_37 + c_4_23·a_1_1·a_1_2·b_1_3
       + c_2_7·a_1_1·a_1_2·b_3_13 + c_4_23·a_1_1·a_1_22 + c_4_22·a_1_23
       + c_4_22·a_1_1·a_1_22 + c_2_7·a_1_1·a_1_2·a_3_10 + c_2_72·a_1_1·a_1_2·b_1_3
  32. b_4_202 + b_4_20·a_1_1·a_1_2·b_1_32 + c_4_22·b_1_34
       + c_4_22·a_1_1·a_1_2·b_1_32 + c_2_72·b_1_34
  33. a_3_9·b_5_37 + c_4_23·a_1_2·a_3_10 + c_4_23·a_1_1·a_3_10 + c_4_23·a_1_0·a_3_10
       + c_4_22·a_1_0·a_3_11 + c_2_7·b_4_20·a_1_1·a_1_2 + c_2_72·a_1_2·a_3_10
       + c_2_72·a_1_1·a_3_10 + c_2_72·a_1_1·a_1_2·b_1_32
  34. b_3_13·b_5_37 + b_4_20·a_1_1·b_3_13 + b_4_20·a_1_1·a_3_10 + c_4_23·b_1_3·b_3_13
       + c_2_7·b_4_20·b_1_32 + c_4_23·a_1_2·b_3_13 + c_4_23·a_1_1·b_3_13
       + c_4_22·a_1_1·b_3_13 + c_2_7·a_1_2·b_1_35 + c_2_7·a_1_1·b_1_32·b_3_13
       + c_2_7·b_4_20·a_1_2·b_1_3 + c_2_7·b_4_20·a_1_1·b_1_3 + c_4_22·a_1_1·a_1_2·b_1_32
       + c_2_7·b_4_20·a_1_1·a_1_2 + c_2_72·b_1_3·b_3_13 + c_2_72·b_1_34
       + c_2_72·a_1_2·b_1_33 + c_2_72·a_1_1·b_3_13 + c_2_72·a_1_1·b_1_33
       + c_2_72·a_1_1·a_1_2·b_1_32
  35. a_3_11·b_5_37 + b_4_20·a_1_1·a_3_10 + c_4_23·b_1_3·a_3_11 + c_4_22·a_1_1·b_3_13
       + c_2_7·b_4_20·a_1_1·b_1_3 + c_4_23·a_1_1·a_3_10 + c_4_23·a_1_0·a_3_11
       + c_4_22·a_1_1·a_1_2·b_1_32 + c_4_22·a_1_0·a_3_10 + c_4_22·a_1_0·a_3_11
       + c_2_7·a_1_1·a_1_2·b_1_3·b_3_13 + c_2_7·a_1_1·a_1_2·b_1_34
       + c_2_7·b_4_20·a_1_1·a_1_2 + c_2_7·a_1_1·a_1_22·a_3_10 + c_2_72·b_1_3·a_3_11
       + c_2_72·a_1_1·b_1_33 + c_2_7·c_4_22·a_1_0·a_1_1 + c_2_72·a_1_1·a_1_2·b_1_32
       + c_2_72·a_1_0·a_3_10 + c_2_72·a_1_0·a_3_11
  36. a_3_10·b_5_37 + c_4_23·b_1_3·a_3_10 + c_4_22·a_1_2·b_3_13 + c_2_7·b_4_20·a_1_2·b_1_3
       + c_2_7·b_4_20·a_1_1·b_1_3 + c_4_23·a_1_2·a_3_10 + c_4_23·a_1_1·a_3_10
       + c_4_22·a_1_1·a_3_10 + c_4_22·a_1_1·a_1_2·b_1_32 + c_2_7·b_4_20·a_1_1·a_1_2
       + c_2_72·b_1_3·a_3_10 + c_2_72·a_1_2·b_1_33 + c_2_72·a_1_1·b_1_33
       + c_2_7·c_4_22·a_1_0·a_1_2 + c_2_72·a_1_1·a_3_10 + c_2_72·a_1_1·a_1_2·b_1_32
  37. b_4_20·b_5_37 + b_4_20·a_1_2·b_1_3·b_3_13 + b_4_20·a_1_1·a_1_2·b_3_13
       + c_4_22·b_1_32·b_3_13 + b_4_20·c_4_23·b_1_3 + c_2_7·b_4_20·b_3_13
       + c_4_22·a_1_2·b_1_3·b_3_13 + c_4_22·a_1_1·b_1_3·b_3_13 + c_4_22·a_1_1·b_1_34
       + b_4_20·c_4_23·a_1_2 + b_4_20·c_4_23·a_1_1 + b_4_20·c_4_22·a_1_1 + c_2_7·b_4_20·a_3_10
       + c_2_7·b_4_20·a_1_1·b_1_32 + c_4_22·a_1_1·a_1_2·b_3_13
       + c_4_22·a_1_1·a_1_2·b_1_33 + c_4_23·a_1_0·a_1_1·a_3_10 + c_4_22·a_1_0·a_1_1·a_3_10
       + c_2_72·b_1_32·b_3_13 + c_2_72·b_4_20·b_1_3 + c_2_7·c_4_22·a_1_2·b_1_32
       + c_2_72·a_1_2·b_1_3·b_3_13 + c_2_72·a_1_1·b_1_3·b_3_13 + c_2_72·a_1_1·b_1_34
       + c_2_72·b_4_20·a_1_2 + c_2_72·b_4_20·a_1_1 + c_2_72·a_1_1·a_1_2·b_1_33
       + c_2_7·c_4_22·a_1_23 + c_2_72·a_1_22·a_3_10 + c_2_72·a_1_0·a_1_1·a_3_10
       + c_2_73·a_1_2·b_1_32 + c_2_73·a_1_23
  38. b_5_372 + c_2_7·a_1_1·a_1_2·b_1_36 + c_2_7·b_4_20·a_1_1·a_1_2·b_1_32
       + c_4_232·b_1_32 + c_2_7·c_4_22·b_1_34 + c_4_232·a_1_22 + c_4_232·a_1_0·a_1_2
       + c_4_222·a_1_0·a_1_2 + c_2_74·b_1_32 + c_2_74·a_1_0·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 10.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_7, a Duflot regular element of degree 2
    2. c_4_22, a Duflot regular element of degree 4
    3. c_4_23, a Duflot regular element of degree 4
    4. b_1_32, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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 3

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. a_3_9 → 0, an element of degree 3
  7. a_3_11 → 0, an element of degree 3
  8. a_3_10 → 0, an element of degree 3
  9. b_3_13 → 0, an element of degree 3
  10. b_4_20 → 0, an element of degree 4
  11. c_4_22 → c_1_14, an element of degree 4
  12. c_4_23 → c_1_04, an element of degree 4
  13. b_5_37 → 0, an element of degree 5

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. a_3_9 → 0, an element of degree 3
  7. a_3_11 → 0, an element of degree 3
  8. a_3_10 → 0, an element of degree 3
  9. b_3_13 → c_1_2·c_1_32, an element of degree 3
  10. b_4_20 → c_1_22·c_1_32 + c_1_12·c_1_32, an element of degree 4
  11. c_4_22 → c_1_14, an element of degree 4
  12. c_4_23 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  13. b_5_37 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_12·c_1_2·c_1_32
       + c_1_02·c_1_33 + c_1_04·c_1_3, an element of degree 5

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