Cohomology of group number 32 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 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • 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.

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

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. 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. b_3_5, an element of degree 3
  8. b_3_6, an element of degree 3
  9. b_4_10, an element of degree 4
  10. c_4_11, a Duflot regular element of degree 4
  11. b_5_17, an element of degree 5

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

Ring relations

There are 33 minimal relations of maximal degree 10:

  1. a_1_02
  2. a_1_0·b_1_1
  3. b_2_1·b_1_1
  4. b_2_2·a_1_0
  5. b_2_3·a_1_0
  6. b_2_1·b_2_2
  7. a_1_0·b_3_4
  8. a_1_0·b_3_5
  9. b_1_1·b_3_5
  10. a_1_0·b_3_6
  11. b_2_1·b_3_4
  12. b_2_2·b_3_5
  13. b_2_1·b_3_6
  14. b_4_10·a_1_0
  15. b_4_10·b_1_1 + b_2_3·b_3_4 + b_2_32·b_1_1 + b_2_2·b_3_6
  16. b_3_4·b_3_5
  17. b_3_52 + b_2_1·b_2_32
  18. b_3_42 + b_1_13·b_3_6 + b_2_2·b_1_1·b_3_4 + b_2_2·b_2_3·b_1_12 + b_2_23
  19. b_3_5·b_3_6
  20. b_3_62 + b_2_3·b_1_1·b_3_6 + b_2_2·b_2_32 + c_4_11·b_1_12
  21. b_2_1·b_4_10 + b_2_1·b_2_32
  22. a_1_0·b_5_17
  23. b_3_4·b_3_6 + b_1_1·b_5_17 + b_2_3·b_1_1·b_3_6 + b_2_3·b_1_1·b_3_4 + b_2_22·b_2_3
  24. b_4_10·b_3_5 + b_2_32·b_3_5
  25. b_2_1·b_5_17
  26. b_4_10·b_3_6 + b_2_3·b_5_17 + b_2_32·b_3_4 + b_2_2·b_2_3·b_3_6 + b_2_2·c_4_11·b_1_1
  27. b_4_10·b_3_4 + b_2_3·b_1_12·b_3_6 + b_2_32·b_3_4 + b_2_2·b_5_17 + b_2_2·b_2_3·b_3_6
       + b_2_2·b_2_32·b_1_1
  28. b_4_102 + b_2_32·b_1_1·b_3_6 + b_2_34 + b_2_2·b_2_3·b_4_10 + b_2_22·c_4_11
  29. b_3_4·b_5_17 + b_2_3·b_1_1·b_5_17 + b_2_32·b_1_1·b_3_6 + b_2_32·b_1_1·b_3_4
       + b_2_2·b_1_1·b_5_17 + b_2_22·b_4_10 + c_4_11·b_1_14
  30. b_3_5·b_5_17
  31. b_3_6·b_5_17 + b_2_32·b_1_1·b_3_6 + b_2_2·b_2_3·b_4_10 + c_4_11·b_1_1·b_3_4
       + b_2_3·c_4_11·b_1_12
  32. b_4_10·b_5_17 + b_2_33·b_3_6 + b_2_33·b_3_4 + b_2_2·b_2_3·b_5_17 + b_2_2·b_2_32·b_3_6
       + b_2_3·c_4_11·b_1_13 + b_2_2·c_4_11·b_3_4 + b_2_2·b_2_3·c_4_11·b_1_1
  33. b_5_172 + b_2_33·b_1_1·b_3_6 + b_2_2·b_2_3·b_1_1·b_5_17 + b_2_2·b_2_32·b_1_1·b_3_6
       + b_2_2·b_2_34 + b_2_22·b_2_3·b_4_10 + c_4_11·b_1_13·b_3_6 + b_2_3·c_4_11·b_1_14
       + b_2_32·c_4_11·b_1_12 + b_2_2·c_4_11·b_1_1·b_3_4 + b_2_2·b_2_3·c_4_11·b_1_12
       + b_2_23·c_4_11

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

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_4_11, a Duflot regular element of degree 4
    2. b_1_1·b_3_6 + b_1_14 + b_2_32 + b_2_22 + b_2_12, an element of degree 4
    3. b_1_13·b_3_6 + b_2_32·b_1_12 + b_2_2·b_1_1·b_3_6 + b_2_2·b_2_32 + b_2_22·b_1_12
         + b_2_1·b_2_32, an element of degree 6
    4. b_1_1, an element of degree 1
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 10, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 3 elements of degree 2.

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

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. b_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. b_3_5 → 0, an element of degree 3
  8. b_3_6 → 0, an element of degree 3
  9. b_4_10 → 0, an element of degree 4
  10. c_4_11 → 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 3

  1. a_1_0 → 0, an element of degree 1
  2. b_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 → 0, an element of degree 3
  7. b_3_5 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_3_6 → 0, an element of degree 3
  9. b_4_10 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  10. c_4_11 → c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2
       + c_1_02·c_1_12 + 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_1, an element of degree 1
  3. b_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  5. b_2_3 → c_1_32 + c_1_2·c_1_3 + c_1_1·c_1_3 + c_1_0·c_1_1, an element of degree 2
  6. b_3_4 → c_1_23 + c_1_1·c_1_2·c_1_3 + c_1_12·c_1_2 + c_1_0·c_1_12, an element of degree 3
  7. b_3_5 → 0, an element of degree 3
  8. b_3_6 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_2·c_1_3 + c_1_02·c_1_1, an element of degree 3
  9. b_4_10 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_23 + c_1_0·c_1_1·c_1_32 + c_1_0·c_1_12·c_1_3
       + c_1_0·c_1_12·c_1_2 + c_1_02·c_1_22 + c_1_02·c_1_1·c_1_2, an element of degree 4
  10. c_4_11 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_1·c_1_2·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_02·c_1_1·c_1_3
       + c_1_02·c_1_1·c_1_2 + c_1_03·c_1_1 + c_1_04, an element of degree 4
  11. b_5_17 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_24·c_1_3
       + c_1_1·c_1_2·c_1_33 + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_2·c_1_32
       + c_1_12·c_1_22·c_1_3 + c_1_13·c_1_2·c_1_3 + c_1_0·c_1_24 + c_1_0·c_1_1·c_1_23
       + c_1_0·c_1_12·c_1_32 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_3
       + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_23 + c_1_02·c_1_1·c_1_32
       + c_1_02·c_1_12·c_1_3 + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13, an element of degree 5

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


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