Cohomology of group number 4 of order 64

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

General information on the group

  • The group has 2 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
    t2  −  t  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)
  • The a-invariants are -∞,-∞,-5,-4,-4. 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 5:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_1, a nilpotent element of degree 2
  4. b_2_2, an element of degree 2
  5. c_2_3, a Duflot regular element of degree 2
  6. a_3_5, a nilpotent element of degree 3
  7. b_3_4, an element of degree 3
  8. b_3_6, an element of degree 3
  9. b_4_9, an element of degree 4
  10. c_4_10, a Duflot regular element of degree 4
  11. b_5_17, an element of degree 5

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Ring relations

There are 33 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_1·a_1_0
  4. a_2_1·b_1_1
  5. b_2_2·a_1_0
  6. a_2_12
  7. a_1_0·a_3_5
  8. b_1_1·a_3_5
  9. a_1_0·b_3_4
  10. a_1_0·b_3_6
  11. a_2_1·a_3_5
  12. a_2_1·b_3_4
  13. b_2_2·a_3_5 + a_2_1·b_3_6
  14. b_4_9·a_1_0
  15. b_4_9·b_1_1 + b_2_2·b_3_4 + b_2_2·a_3_5
  16. a_3_52
  17. a_3_5·b_3_4
  18. b_3_42 + c_2_3·b_1_14
  19. a_3_5·b_3_6 + a_2_1·b_2_22
  20. b_3_62 + b_2_2·b_1_1·b_3_6 + b_2_2·b_1_1·b_3_4 + b_2_23 + c_4_10·b_1_12
       + b_2_2·c_2_3·b_1_12
  21. a_2_1·b_4_9
  22. a_1_0·b_5_17
  23. b_3_4·b_3_6 + b_1_1·b_5_17 + b_2_2·b_1_1·b_3_6 + b_2_2·b_1_1·b_3_4 + a_2_1·b_2_22
       + c_2_3·b_1_1·b_3_4 + c_2_3·b_1_14
  24. b_4_9·a_3_5
  25. b_4_9·b_3_4 + b_2_2·c_2_3·b_1_13
  26. b_4_9·b_3_6 + b_2_2·b_5_17 + b_2_22·b_3_6 + b_2_22·b_3_4 + a_2_1·b_2_2·b_3_6
       + b_2_2·c_2_3·b_3_4 + b_2_2·c_2_3·b_1_13 + a_2_1·c_2_3·b_3_6
  27. a_2_1·b_5_17 + a_2_1·b_2_2·b_3_6
  28. b_4_92 + b_2_22·c_2_3·b_1_12
  29. a_3_5·b_5_17 + a_2_1·b_2_23
  30. b_3_4·b_5_17 + b_2_2·b_1_1·b_5_17 + b_2_22·b_1_1·b_3_6 + b_2_22·b_1_1·b_3_4
       + a_2_1·b_2_23 + c_2_3·b_1_13·b_3_6 + c_2_3·b_1_13·b_3_4 + b_2_2·c_2_3·b_1_1·b_3_4
       + c_2_32·b_1_14
  31. b_3_6·b_5_17 + b_2_22·b_1_1·b_3_6 + b_2_22·b_1_1·b_3_4 + b_2_22·b_4_9 + b_2_24
       + c_4_10·b_1_1·b_3_4 + c_2_3·b_1_1·b_5_17 + c_2_3·b_1_13·b_3_6 + b_2_2·c_4_10·b_1_12
       + b_2_2·c_2_3·b_1_1·b_3_6 + b_2_2·c_2_3·b_1_14 + b_2_22·c_2_3·b_1_12
       + c_2_32·b_1_1·b_3_4 + c_2_32·b_1_14
  32. b_4_9·b_5_17 + b_2_22·b_5_17 + b_2_23·b_3_6 + b_2_23·b_3_4 + a_2_1·b_2_22·b_3_6
       + b_2_2·c_2_3·b_1_12·b_3_6 + b_2_2·c_2_3·b_1_12·b_3_4 + b_2_22·c_2_3·b_3_4
       + a_2_1·b_2_2·c_2_3·b_3_6 + b_2_2·c_2_32·b_1_13
  33. b_5_172 + b_2_23·b_1_1·b_3_6 + b_2_23·b_1_1·b_3_4 + b_2_25
       + b_2_2·c_2_3·b_1_13·b_3_6 + b_2_2·c_2_3·b_1_13·b_3_4 + b_2_22·c_4_10·b_1_12
       + b_2_22·c_2_3·b_1_14 + c_2_3·c_4_10·b_1_14 + c_2_32·b_1_16
       + b_2_2·c_2_32·b_1_14 + c_2_33·b_1_14

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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_3, a Duflot regular element of degree 2
    2. c_4_10, a Duflot regular element of degree 4
    3. b_1_12 + b_2_2, an element of degree 2
    4. b_1_1, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 1, 4, 5].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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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. a_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. c_2_3 → c_1_12, an element of degree 2
  6. a_3_5 → 0, an element of degree 3
  7. b_3_4 → 0, an element of degree 3
  8. b_3_6 → 0, an element of degree 3
  9. b_4_9 → 0, an element of degree 4
  10. c_4_10 → c_1_04, an element of degree 4
  11. b_5_17 → 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. b_1_1 → c_1_2, an element of degree 1
  3. a_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  5. c_2_3 → c_1_12, an element of degree 2
  6. a_3_5 → 0, an element of degree 3
  7. b_3_4 → c_1_1·c_1_22, an element of degree 3
  8. b_3_6 → c_1_33 + c_1_22·c_1_3 + c_1_1·c_1_22 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  9. b_4_9 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3, an element of degree 4
  10. 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_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
  11. b_5_17 → c_1_35 + c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_1·c_1_2·c_1_33
       + c_1_1·c_1_23·c_1_3 + c_1_13·c_1_22 + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_23·c_1_3
       + c_1_0·c_1_1·c_1_23 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3
       + c_1_02·c_1_1·c_1_22, an element of degree 5

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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