Cohomology of group number 391 of order 128

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General information on the group

  • The group has 3 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • 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 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t6  −  t5  +  t4  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-3,-5,-4. They were obtained using the first, the second, the second power of the third, and the fourth filter regular parameter of the Benson test.

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Ring generators

The cohomology ring has 11 minimal generators of maximal degree 6:

  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. a_2_4, a nilpotent element of degree 2
  5. a_3_5, a nilpotent element of degree 3
  6. b_3_6, an element of degree 3
  7. b_4_8, an element of degree 4
  8. b_4_9, an element of degree 4
  9. c_4_10, a Duflot regular element of degree 4
  10. c_4_11, a Duflot regular element of degree 4
  11. b_6_25, an element of degree 6

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Ring relations

There are 33 minimal relations of maximal degree 12:

  1. a_1_0·b_1_1
  2. a_1_0·b_1_2
  3. a_2_4·b_1_2
  4. a_2_4·a_1_0
  5. a_2_4·b_1_1 + a_1_03
  6. a_2_42
  7. b_1_2·a_3_5
  8. b_1_1·a_3_5
  9. a_1_0·b_3_6 + a_1_0·a_3_5
  10. a_2_4·a_3_5
  11. a_2_4·b_3_6 + a_1_02·a_3_5
  12. b_4_8·a_1_0 + a_1_02·a_3_5
  13. b_1_1·b_1_2·b_3_6 + b_1_12·b_3_6 + b_4_9·b_1_2 + b_4_8·b_1_2 + b_4_8·b_1_1
       + a_1_02·a_3_5
  14. b_4_9·a_1_0
  15. a_3_5·b_3_6 + a_3_52
  16. a_3_52 + c_4_10·a_1_02
  17. a_2_4·b_4_8
  18. a_2_4·b_4_9
  19. b_3_62 + b_1_23·b_3_6 + b_1_13·b_3_6 + b_4_9·b_1_22 + b_4_9·b_1_12
       + b_4_8·b_1_1·b_1_2 + a_3_52 + c_4_11·b_1_22 + c_4_10·b_1_12
  20. b_4_8·a_3_5 + c_4_10·a_1_03
  21. b_4_9·a_3_5
  22. b_1_14·b_3_6 + b_6_25·b_1_2 + b_4_9·b_1_23 + b_4_9·b_1_13 + b_4_8·b_3_6
       + b_4_8·b_1_1·b_1_22 + b_4_8·b_1_12·b_1_2 + c_4_11·b_1_1·b_1_22
       + c_4_10·b_1_1·b_1_22 + c_4_10·b_1_13
  23. b_6_25·a_1_0 + c_4_10·a_1_03
  24. b_1_14·b_3_6 + b_6_25·b_1_1 + b_4_9·b_3_6 + b_4_9·b_1_23 + b_4_9·b_1_12·b_1_2
       + b_4_9·b_1_13 + b_4_8·b_3_6 + b_4_8·b_1_23 + b_4_8·b_1_1·b_1_22
       + c_4_11·b_1_1·b_1_22 + c_4_10·b_1_12·b_1_2 + c_4_10·b_1_13 + c_4_11·a_1_03
       + c_4_10·a_1_03
  25. b_1_25·b_3_6 + b_4_9·b_1_13·b_1_2 + b_4_9·b_1_14 + b_4_8·b_1_12·b_1_22
       + b_4_8·b_1_13·b_1_2 + b_4_8·b_1_14 + b_4_82 + c_4_11·b_1_24 + c_4_10·b_1_24
       + c_4_10·b_1_14
  26. b_4_9·b_1_24 + b_4_9·b_1_1·b_3_6 + b_4_9·b_1_12·b_1_22 + b_4_9·b_1_13·b_1_2
       + b_4_92 + b_4_8·b_1_24 + b_4_8·b_1_1·b_1_23 + b_4_8·b_1_12·b_1_22
       + b_4_8·b_1_14 + b_4_8·b_4_9 + c_4_11·b_1_1·b_1_23 + c_4_11·b_1_12·b_1_22
       + c_4_11·b_1_13·b_1_2 + c_4_11·b_1_14 + c_4_10·b_1_1·b_1_23
       + c_4_10·b_1_12·b_1_22
  27. a_2_4·b_6_25
  28. b_6_25·b_1_12 + b_4_9·b_1_1·b_3_6 + b_4_9·b_1_1·b_1_23 + b_4_9·b_1_12·b_1_22
       + b_4_92 + b_4_8·b_1_1·b_3_6 + b_4_8·b_1_1·b_1_23 + b_4_8·b_1_12·b_1_22
       + b_4_8·b_1_14 + b_4_82 + c_4_11·b_1_12·b_1_22 + c_4_11·b_1_14
       + c_4_10·b_1_12·b_1_22 + c_4_10·b_1_13·b_1_2
  29. b_6_25·a_3_5 + c_4_10·a_1_02·a_3_5
  30. b_6_25·b_3_6 + b_4_92·b_1_2 + b_4_92·b_1_1 + b_4_8·b_1_1·b_1_24
       + b_4_8·b_1_13·b_1_22 + b_4_8·b_1_14·b_1_2 + b_4_8·b_4_9·b_1_2 + b_4_8·b_4_9·b_1_1
       + b_4_82·b_1_2 + c_4_11·b_1_12·b_3_6 + c_4_11·b_1_12·b_1_23
       + c_4_11·b_1_13·b_1_22 + c_4_11·b_1_14·b_1_2 + c_4_10·b_1_1·b_1_24
       + c_4_10·b_1_12·b_1_23 + c_4_10·b_1_13·b_1_22 + b_4_9·c_4_11·b_1_2
       + b_4_9·c_4_10·b_1_2 + b_4_9·c_4_10·b_1_1 + b_4_8·c_4_11·b_1_1 + b_4_8·c_4_10·b_1_2
  31. b_4_9·b_6_25 + b_4_92·b_1_22 + b_4_92·b_1_1·b_1_2 + b_4_92·b_1_12
       + b_4_8·b_1_26 + b_4_8·b_1_1·b_1_25 + b_4_8·b_1_13·b_1_23 + b_4_8·b_1_16
       + b_4_8·b_4_9·b_1_22 + b_4_8·b_4_9·b_1_1·b_1_2 + b_4_82·b_1_1·b_1_2
       + b_4_82·b_1_12 + c_4_11·b_1_23·b_3_6 + c_4_11·b_1_1·b_1_25
       + c_4_11·b_1_13·b_3_6 + c_4_11·b_1_13·b_1_23 + c_4_11·b_1_14·b_1_22
       + c_4_11·b_1_16 + c_4_10·b_1_23·b_3_6 + c_4_10·b_1_26 + c_4_10·b_1_12·b_1_24
       + c_4_10·b_1_13·b_3_6 + c_4_10·b_1_13·b_1_23 + c_4_10·b_1_15·b_1_2
       + b_4_9·c_4_11·b_1_22 + b_4_9·c_4_10·b_1_22 + b_4_8·c_4_11·b_1_22
       + b_4_8·c_4_11·b_1_1·b_1_2 + b_4_8·c_4_10·b_1_22 + b_4_8·c_4_10·b_1_12
  32. b_4_8·b_1_26 + b_4_8·b_1_14·b_1_22 + b_4_8·b_1_16 + b_4_8·b_6_25
       + b_4_8·b_4_9·b_1_22 + b_4_8·b_4_9·b_1_1·b_1_2 + b_4_8·b_4_9·b_1_12
       + b_4_82·b_1_22 + b_4_82·b_1_1·b_1_2 + c_4_11·b_1_23·b_3_6 + c_4_11·b_1_1·b_1_25
       + c_4_11·b_1_12·b_1_24 + c_4_11·b_1_16 + c_4_10·b_1_23·b_3_6 + c_4_10·b_1_26
       + c_4_10·b_1_1·b_1_25 + c_4_10·b_1_13·b_3_6 + c_4_10·b_1_15·b_1_2
       + b_4_9·c_4_10·b_1_12 + b_4_8·c_4_11·b_1_1·b_1_2 + b_4_8·c_4_10·b_1_1·b_1_2
       + b_4_8·c_4_10·b_1_12
  33. b_6_252 + b_4_92·b_1_13·b_1_2 + b_4_92·b_1_14 + b_4_8·b_1_15·b_1_23
       + b_4_8·b_1_17·b_1_2 + b_4_8·b_1_18 + b_4_8·b_4_9·b_1_1·b_1_23
       + b_4_8·b_4_9·b_1_13·b_1_2 + b_4_8·b_4_9·b_1_14 + b_4_82·b_1_2·b_3_6
       + b_4_82·b_1_24 + b_4_82·b_1_1·b_3_6 + b_4_82·b_1_1·b_1_23
       + b_4_82·b_1_12·b_1_22 + b_4_82·b_1_14 + b_4_83 + c_4_11·b_1_16·b_1_22
       + c_4_11·b_1_17·b_1_2 + c_4_11·b_1_18 + c_4_10·b_1_15·b_1_23
       + c_4_10·b_1_16·b_1_22 + c_4_10·b_1_17·b_1_2 + b_4_9·c_4_11·b_1_1·b_1_23
       + b_4_9·c_4_11·b_1_12·b_1_22 + b_4_9·c_4_10·b_1_12·b_1_22
       + b_4_9·c_4_10·b_1_14 + b_4_92·c_4_10 + b_4_8·c_4_11·b_1_1·b_1_23
       + b_4_8·c_4_11·b_1_12·b_1_22 + b_4_8·c_4_10·b_1_12·b_1_22
       + b_4_8·c_4_10·b_1_14 + b_4_82·c_4_11 + b_4_82·c_4_10 + c_4_112·b_1_12·b_1_22
       + c_4_10·c_4_11·b_1_12·b_1_22 + c_4_10·c_4_11·b_1_14 + c_4_102·b_1_12·b_1_22
       + c_4_102·b_1_14

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 12.
  • 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_4_10, a Duflot regular element of degree 4
    2. c_4_11, a Duflot regular element of degree 4
    3. b_1_22 + b_1_1·b_1_2 + b_1_12, an element of degree 2
    4. b_1_22, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 5, 5, 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 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. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. b_3_6 → 0, an element of degree 3
  7. b_4_8 → 0, an element of degree 4
  8. b_4_9 → 0, an element of degree 4
  9. c_4_10 → c_1_14, an element of degree 4
  10. c_4_11 → c_1_14 + c_1_04, an element of degree 4
  11. b_6_25 → 0, an element of degree 6

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 → c_1_2, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. b_3_6 → c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_32
       + c_1_02·c_1_3, an element of degree 3
  7. b_4_8 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_22
       + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32, an element of degree 4
  8. b_4_9 → c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22
       + c_1_0·c_1_33 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23
       + c_1_02·c_1_32 + c_1_02·c_1_22, an element of degree 4
  9. c_4_10 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_33
       + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3
       + c_1_02·c_1_22, an element of degree 4
  10. c_4_11 → c_1_1·c_1_33 + c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_14 + 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
  11. b_6_25 → c_1_1·c_1_2·c_1_34 + c_1_12·c_1_2·c_1_33 + c_1_12·c_1_22·c_1_32
       + c_1_12·c_1_23·c_1_3 + c_1_12·c_1_24 + c_1_13·c_1_2·c_1_32
       + c_1_13·c_1_22·c_1_3 + c_1_14·c_1_22 + c_1_0·c_1_35 + c_1_0·c_1_2·c_1_34
       + c_1_0·c_1_22·c_1_33 + c_1_0·c_1_1·c_1_34 + c_1_0·c_1_1·c_1_22·c_1_32
       + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_33 + c_1_0·c_1_12·c_1_22·c_1_3
       + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_1·c_1_33
       + c_1_02·c_1_1·c_1_2·c_1_32 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_32
       + c_1_02·c_1_12·c_1_2·c_1_3 + c_1_02·c_1_12·c_1_22 + c_1_03·c_1_2·c_1_32
       + c_1_03·c_1_22·c_1_3 + c_1_04·c_1_32 + c_1_04·c_1_2·c_1_3, an element of degree 6

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