Cohomology of group number 898 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 16.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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
    ( − 1) · (t4  −  t2  +  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 11 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent element of degree 1
  3. b_1_1, an element of degree 1
  4. a_3_3, a nilpotent element of degree 3
  5. b_4_4, an element of degree 4
  6. a_5_4, a nilpotent element of degree 5
  7. b_5_6, an element of degree 5
  8. b_6_8, an element of degree 6
  9. b_7_10, an element of degree 7
  10. c_8_13, a Duflot regular element of degree 8
  11. b_9_17, an element of degree 9

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Ring relations

There are 34 minimal relations of maximal degree 18:

  1. a_1_02
  2. a_1_0·b_1_1 + a_1_22
  3. a_1_22·b_1_1
  4. a_1_0·a_3_3
  5. b_1_12·a_3_3 + a_1_2·b_1_14
  6. b_4_4·a_1_0
  7. a_3_32
  8. a_1_0·a_5_4
  9. a_1_0·b_5_6
  10. b_1_12·a_5_4
  11. b_6_8·a_1_0
  12. a_3_3·a_5_4
  13. a_3_3·b_5_6 + a_1_2·b_1_12·b_5_6
  14. a_1_0·b_7_10
  15. b_6_8·a_3_3 + b_6_8·a_1_2·b_1_12 + b_4_4·a_5_4
  16. b_1_12·b_7_10 + b_1_14·b_5_6 + b_6_8·b_1_13 + b_4_4·b_5_6 + b_4_4·a_5_4
  17. a_5_42
  18. a_5_4·b_5_6
  19. b_5_62 + b_4_4·b_1_16 + a_1_2·b_1_19
  20. a_3_3·b_7_10 + a_1_2·b_1_14·b_5_6 + b_6_8·a_1_2·b_1_13 + b_4_4·a_1_2·b_5_6
  21. a_1_0·b_9_17
  22. b_6_8·a_5_4 + b_4_42·a_3_3 + b_4_42·a_1_2·b_1_12
  23. b_1_12·b_9_17 + b_6_8·b_5_6 + b_4_42·b_1_13 + a_1_2·b_1_110 + b_4_4·a_1_2·b_1_16
       + b_4_42·a_3_3 + b_4_42·a_1_2·b_1_12
  24. a_5_4·b_7_10 + b_4_42·a_1_2·a_3_3
  25. b_5_6·b_7_10 + b_6_8·b_1_1·b_5_6 + b_4_4·b_1_18 + b_4_42·b_1_14 + a_1_2·b_1_111
       + b_4_4·a_1_2·b_1_17 + b_4_42·b_1_1·a_3_3 + b_4_42·a_1_2·b_1_13
       + b_4_42·a_1_2·a_3_3
  26. b_6_8·b_1_16 + b_6_82 + b_4_4·b_1_13·b_5_6 + b_4_4·b_1_18 + b_4_4·b_6_8·b_1_12
       + b_4_43 + a_1_2·b_1_16·b_5_6 + b_4_4·a_1_2·b_1_17 + b_4_42·b_1_1·a_3_3
       + c_8_13·b_1_14
  27. a_3_3·b_9_17 + b_6_8·a_1_2·b_5_6 + b_4_42·a_1_2·b_1_13
  28. b_6_8·b_7_10 + b_6_8·b_1_12·b_5_6 + b_6_82·b_1_1 + b_4_4·b_9_17 + b_4_43·b_1_1
       + b_4_4·a_1_2·b_1_18 + b_4_42·a_1_2·b_1_14 + b_4_42·a_1_2·b_1_1·a_3_3
  29. b_7_102 + b_6_82·b_1_12 + b_4_4·b_1_110 + b_4_43·b_1_12 + a_1_2·b_1_113
       + b_4_42·a_1_2·b_1_15
  30. a_5_4·b_9_17 + b_4_42·a_1_2·a_5_4
  31. b_5_6·b_9_17 + b_4_4·b_6_8·b_1_14 + b_4_42·b_1_1·b_5_6 + a_1_2·b_1_18·b_5_6
       + b_6_82·a_1_2·b_1_1 + b_4_4·a_1_2·b_1_19 + b_4_4·b_6_8·a_1_2·b_1_13
       + b_4_42·b_1_1·a_5_4 + b_4_43·a_1_2·b_1_1 + b_4_42·a_1_2·a_5_4
       + c_8_13·a_1_2·b_1_15
  32. b_6_8·b_9_17 + b_6_8·b_1_14·b_5_6 + b_4_4·b_1_16·b_5_6 + b_4_4·b_6_8·b_5_6
       + b_4_42·b_7_10 + b_4_42·b_1_12·b_5_6 + b_4_42·b_1_17 + b_6_82·a_1_2·b_1_12
       + b_4_4·a_1_2·b_1_110 + b_4_42·a_1_2·b_1_1·b_5_6 + b_4_43·a_3_3
       + b_4_42·a_1_2·b_1_1·a_5_4 + c_8_13·b_1_12·b_5_6 + c_8_13·a_1_2·b_1_16
  33. b_7_10·b_9_17 + b_6_8·b_1_15·b_5_6 + b_4_4·b_1_17·b_5_6 + b_4_4·b_6_8·b_1_1·b_5_6
       + b_4_4·b_6_82 + b_4_42·b_6_8·b_1_12 + b_4_44 + a_1_2·b_1_110·b_5_6
       + b_4_42·a_1_2·b_1_17 + b_4_43·a_1_2·b_1_13 + c_8_13·b_1_13·b_5_6
       + b_4_4·c_8_13·b_1_14
  34. b_9_172 + b_4_4·b_6_82·b_1_12 + b_4_44·b_1_12 + b_6_82·a_1_2·b_1_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 18.
  • 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_13, a Duflot regular element of degree 8
    2. b_1_14 + b_4_4, an element of degree 4
    3. b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, 3, 9, 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_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. b_4_4 → 0, an element of degree 4
  6. a_5_4 → 0, an element of degree 5
  7. b_5_6 → 0, an element of degree 5
  8. b_6_8 → 0, an element of degree 6
  9. b_7_10 → 0, an element of degree 7
  10. c_8_13 → c_1_08, an element of degree 8
  11. b_9_17 → 0, an element of degree 9

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → c_1_1, an element of degree 1
  4. a_3_3 → 0, an element of degree 3
  5. b_4_4 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  6. a_5_4 → 0, an element of degree 5
  7. b_5_6 → c_1_13·c_1_22 + c_1_14·c_1_2, an element of degree 5
  8. b_6_8 → c_1_26 + c_1_15·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  9. b_7_10 → c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23 + c_1_15·c_1_22
       + c_1_02·c_1_15 + c_1_04·c_1_13, an element of degree 7
  10. c_8_13 → c_1_12·c_1_26 + c_1_17·c_1_2 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
       + 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
  11. b_9_17 → c_1_12·c_1_27 + c_1_15·c_1_24 + c_1_16·c_1_23 + c_1_17·c_1_22
       + c_1_02·c_1_15·c_1_22 + c_1_02·c_1_16·c_1_2 + c_1_04·c_1_13·c_1_22
       + c_1_04·c_1_14·c_1_2, an element of degree 9

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