Cohomology of group number 136 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 of rank 2 and 3, respectively.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    (t2  −  t  +  1) · (t5  +  t4  −  t2  −  t  −  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. b_1_1, an element of degree 1
  3. a_2_2, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. b_2_3, an element of degree 2
  6. b_3_5, an element of degree 3
  7. a_5_7, a nilpotent element of degree 5
  8. b_6_10, an element of degree 6
  9. c_8_14, a Duflot regular element of degree 8

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Ring relations

There are 22 minimal relations of maximal degree 12:

  1. a_1_02
  2. a_1_0·b_1_1
  3. a_2_2·a_1_0
  4. b_2_1·b_1_1
  5. a_2_2·b_2_1
  6. a_1_0·b_3_5
  7. b_1_1·b_3_5 + b_2_3·b_1_12 + a_2_22
  8. b_2_1·b_2_3·a_1_0
  9. b_2_32·b_1_1 + b_2_32·a_1_0 + a_2_2·b_2_3·b_1_1 + a_2_22·b_1_1
  10. a_2_2·b_3_5 + a_2_2·b_2_3·b_1_1
  11. b_3_52 + b_2_1·b_2_32 + a_2_2·b_2_3·b_1_12 + a_2_2·b_2_32 + a_2_22·b_1_12
       + a_2_22·b_2_3
  12. a_1_0·a_5_7
  13. b_1_1·a_5_7 + a_2_2·b_2_3·b_1_12 + a_2_22·b_2_3
  14. a_2_2·a_5_7 + a_2_22·b_2_3·b_1_1
  15. b_6_10·a_1_0 + b_2_1·a_5_7
  16. b_6_10·b_1_1 + a_2_22·b_1_13 + a_2_22·b_2_3·b_1_1
  17. b_3_5·a_5_7 + a_2_22·b_2_3·b_1_12
  18. a_2_2·b_6_10
  19. b_2_1·b_2_3·a_5_7
  20. a_5_72
  21. b_6_10·a_5_7 + b_2_13·a_5_7 + b_2_1·c_8_14·a_1_0
  22. b_6_102 + b_2_12·b_2_3·b_6_10 + b_2_12·b_2_34 + b_2_13·b_6_10 + b_2_13·b_2_33
       + b_2_14·b_2_32 + b_2_12·c_8_14

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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_8_14, a Duflot regular element of degree 8
    2. b_1_12 + b_2_3 + b_2_1, an element of degree 2
    3. b_3_5, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 10].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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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. a_2_2 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_3_5 → 0, an element of degree 3
  7. a_5_7 → 0, an element of degree 5
  8. b_6_10 → 0, an element of degree 6
  9. c_8_14 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_1, an element of degree 1
  3. a_2_2 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_3 → 0, an element of degree 2
  6. b_3_5 → 0, an element of degree 3
  7. a_5_7 → 0, an element of degree 5
  8. b_6_10 → 0, an element of degree 6
  9. c_8_14 → 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. b_1_1 → 0, an element of degree 1
  3. a_2_2 → 0, an element of degree 2
  4. b_2_1 → c_1_12, 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_5 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. a_5_7 → 0, an element of degree 5
  8. b_6_10 → c_1_14·c_1_22 + 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
  9. c_8_14 → 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_16·c_1_2 + c_1_02·c_1_16
       + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_08, an element of degree 8

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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