Cohomology of group number 141 of order 128

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General information on the group

  • The group has 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has 2 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t7  −  t6  −  t5  +  t3  −  t2  −  1
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-5,-3,-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_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_2_1, an element of degree 2
  4. b_2_2, an element of degree 2
  5. b_2_3, an element of degree 2
  6. b_3_4, an element of degree 3
  7. a_5_3, a nilpotent element of degree 5
  8. b_5_7, an element of degree 5
  9. b_6_9, an element of degree 6
  10. b_6_10, an element of degree 6
  11. b_7_12, an element of degree 7
  12. c_8_15, a Duflot regular element of degree 8

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Ring relations

There are 44 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_0·a_1_1
  3. b_2_2·a_1_1 + b_2_1·a_1_1
  4. b_2_2·a_1_0 + b_2_1·a_1_1
  5. b_2_3·a_1_0 + a_1_13
  6. b_2_22 + b_2_1·b_2_2
  7. a_1_1·b_3_4
  8. a_1_0·b_3_4
  9. b_2_3·a_1_13
  10. b_3_42 + b_2_1·b_2_32
  11. a_1_1·a_5_3 + b_2_32·a_1_12
  12. a_1_0·a_5_3
  13. a_1_0·b_5_7
  14. b_2_2·b_2_3·b_3_4 + b_2_1·b_5_7 + b_2_1·b_2_3·b_3_4 + b_2_1·b_2_2·b_3_4 + b_2_2·a_5_3
  15. b_2_2·b_5_7 + b_2_1·b_2_2·b_3_4 + b_2_2·a_5_3
  16. b_2_3·a_5_3 + b_2_33·a_1_1 + a_1_12·b_5_7
  17. b_6_9·a_1_1 + b_2_3·a_5_3 + b_2_33·a_1_1 + b_2_13·a_1_1
  18. b_6_9·a_1_0 + b_2_2·a_5_3 + b_2_1·a_5_3 + b_2_13·a_1_1
  19. b_6_10·a_1_1 + b_2_33·a_1_1 + b_2_2·a_5_3
  20. b_6_10·a_1_0 + b_2_1·a_5_3
  21. b_3_4·a_5_3
  22. b_3_4·b_5_7 + b_2_2·b_2_33 + b_2_1·b_2_33 + b_2_1·b_2_2·b_2_32
  23. b_2_2·b_6_9 + b_2_13·b_2_2
  24. b_2_2·b_6_10 + b_2_2·b_2_33 + b_2_1·b_6_10 + b_2_1·b_6_9 + b_2_1·b_2_33
       + b_2_12·b_2_2·b_2_3 + b_2_13·b_2_3 + b_2_13·b_2_2
  25. a_1_1·b_7_12
  26. a_1_0·b_7_12
  27. b_6_10·b_3_4 + b_2_33·b_3_4 + b_2_1·b_7_12 + b_2_1·b_2_3·b_5_7 + b_2_12·b_5_7
       + b_2_12·b_2_2·b_3_4 + b_2_13·b_3_4 + b_2_1·b_2_2·a_5_3
  28. b_6_10·b_3_4 + b_6_9·b_3_4 + b_2_33·b_3_4 + b_2_2·b_7_12 + b_2_12·b_2_3·b_3_4
  29. a_5_32 + b_2_34·a_1_12
  30. a_5_3·b_5_7 + b_2_32·a_1_1·b_5_7
  31. b_3_4·b_7_12 + b_2_32·b_6_10 + b_2_35 + b_2_2·b_2_34 + b_2_1·b_2_34
       + b_2_12·b_2_33 + b_2_13·b_2_32
  32. b_5_72 + b_2_2·b_2_34 + b_2_1·b_2_34 + b_2_12·b_2_2·b_2_32 + a_5_3·b_5_7
       + b_2_34·a_1_12 + c_8_15·a_1_12
  33. b_6_10·b_5_7 + b_2_33·b_5_7 + b_2_2·b_2_3·b_7_12 + b_2_1·b_2_3·b_7_12
       + b_2_1·b_2_32·b_5_7 + b_2_1·b_2_2·b_7_12 + b_2_12·b_2_32·b_3_4
       + b_2_13·b_2_3·b_3_4 + b_2_13·b_2_2·b_3_4 + b_6_10·a_5_3 + b_6_9·a_5_3 + b_2_35·a_1_1
       + b_2_12·b_2_2·a_5_3
  34. b_6_9·b_5_7 + b_2_2·b_2_3·b_7_12 + b_2_1·b_2_3·b_7_12 + b_2_1·b_2_32·b_5_7
       + b_2_12·b_2_3·b_5_7 + b_2_12·b_2_32·b_3_4 + b_2_13·b_2_3·b_3_4
       + b_2_13·b_2_2·b_3_4 + b_2_12·b_2_2·a_5_3 + b_2_32·a_1_12·b_5_7 + c_8_15·a_1_13
  35. b_6_10·a_5_3 + b_6_9·a_5_3 + b_2_35·a_1_1 + b_2_1·c_8_15·a_1_1
  36. b_6_10·a_5_3 + b_2_35·a_1_1 + b_2_12·b_2_2·a_5_3 + b_2_32·a_1_12·b_5_7
       + b_2_1·c_8_15·a_1_0
  37. b_6_9·b_6_10 + b_6_92 + b_2_33·b_6_9 + b_2_12·b_2_3·b_6_9 + b_2_13·b_6_10
       + b_2_13·b_6_9 + b_2_13·b_2_33 + b_2_15·b_2_3
  38. a_5_3·b_7_12
  39. b_5_7·b_7_12 + b_2_33·b_6_9 + b_2_2·b_2_35 + b_2_1·b_2_32·b_6_10
       + b_2_1·b_2_32·b_6_9 + b_2_33·a_1_1·b_5_7
  40. b_6_102 + b_6_9·b_6_10 + b_6_92 + b_2_33·b_6_9 + b_2_36 + b_2_1·b_2_2·b_2_34
       + b_2_12·b_2_2·b_2_33 + b_2_13·b_2_2·b_2_32 + b_2_14·b_2_2·b_2_3 + b_2_15·b_2_3
       + b_2_15·b_2_2 + b_2_12·c_8_15
  41. b_6_102 + b_6_9·b_6_10 + b_2_33·b_6_9 + b_2_36 + b_2_1·b_2_2·b_2_34
       + b_2_12·b_2_3·b_6_9 + b_2_12·b_2_2·b_2_33 + b_2_14·b_2_32 + b_2_14·b_2_2·b_2_3
       + b_2_1·b_2_2·c_8_15
  42. b_6_10·b_7_12 + b_2_33·b_7_12 + b_2_2·b_2_32·b_7_12 + b_2_1·b_2_32·b_7_12
       + b_2_1·b_2_34·b_3_4 + b_2_1·b_2_2·b_2_3·b_7_12 + b_2_12·b_2_33·b_3_4
       + b_2_12·b_2_2·b_7_12 + b_2_13·b_7_12 + b_2_13·b_2_3·b_5_7 + b_2_14·b_2_2·b_3_4
       + b_2_15·b_3_4 + b_2_1·c_8_15·b_3_4
  43. b_6_10·b_7_12 + b_6_9·b_7_12 + b_2_33·b_7_12 + b_2_1·b_2_33·b_5_7
       + b_2_1·b_2_34·b_3_4 + b_2_12·b_2_3·b_7_12 + b_2_12·b_2_2·b_7_12 + b_2_14·b_5_7
       + b_2_14·b_2_3·b_3_4 + b_2_14·b_2_2·b_3_4 + b_2_13·b_2_2·a_5_3 + b_2_2·c_8_15·b_3_4
  44. b_7_122 + b_2_1·b_2_33·b_6_9 + b_2_1·b_2_36 + b_2_1·b_2_2·b_2_35
       + b_2_12·b_2_32·b_6_10 + b_2_12·b_2_32·b_6_9 + b_2_12·b_2_35
       + b_2_12·b_2_2·b_2_34 + b_2_13·b_2_34 + b_2_13·b_2_2·b_2_33
       + b_2_14·b_2_2·b_2_32 + b_2_15·b_2_32 + b_2_1·b_2_32·c_8_15

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 14.
  • 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_8_15, a Duflot regular element of degree 8
    2. b_2_3 + b_2_1, an element of degree 2
    3. b_3_4, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 7, 10].
  • 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_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_3_4 → 0, an element of degree 3
  7. a_5_3 → 0, an element of degree 5
  8. b_5_7 → 0, an element of degree 5
  9. b_6_9 → 0, an element of degree 6
  10. b_6_10 → 0, an element of degree 6
  11. b_7_12 → 0, an element of degree 7
  12. c_8_15 → c_1_08, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_2_1 → c_1_12, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_2_3 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  6. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. a_5_3 → 0, an element of degree 5
  8. b_5_7 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_6_9 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14
       + c_1_04·c_1_12, an element of degree 6
  10. b_6_10 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_0·c_1_13·c_1_22
       + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2
       + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  11. b_7_12 → c_1_15·c_1_22 + c_1_16·c_1_2 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22
       + c_1_04·c_1_12·c_1_2, an element of degree 7
  12. c_8_15 → c_1_28 + 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_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_24
       + c_1_04·c_1_13·c_1_2 + 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_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. b_2_1 → c_1_22, an element of degree 2
  4. b_2_2 → c_1_22, an element of degree 2
  5. b_2_3 → c_1_1·c_1_2 + c_1_12, an element of degree 2
  6. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. a_5_3 → 0, an element of degree 5
  8. b_5_7 → c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  9. b_6_9 → c_1_26, an element of degree 6
  10. b_6_10 → c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_16 + c_1_0·c_1_1·c_1_24
       + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23
       + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22, an element of degree 6
  11. b_7_12 → c_1_1·c_1_26 + c_1_14·c_1_23 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22
       + c_1_04·c_1_12·c_1_2, an element of degree 7
  12. c_8_15 → 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_16·c_1_22 + c_1_18 + c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25
       + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22
       + 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