Cohomology of group number 1858 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 2.
  • It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 3 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t8  +  t5  +  t4  +  t3  +  t2  +  t  +  1)
    (t  −  1)3 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.

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Ring generators

The cohomology ring has 10 minimal generators of maximal degree 8:

  1. a_1_2, a nilpotent element of degree 1
  2. b_1_0, an element of degree 1
  3. b_1_1, an element of degree 1
  4. b_1_3, an element of degree 1
  5. a_3_10, a nilpotent element of degree 3
  6. b_3_11, an element of degree 3
  7. c_4_16, a Duflot regular element of degree 4
  8. b_5_22, an element of degree 5
  9. b_5_23, an element of degree 5
  10. c_8_42, a Duflot regular element of degree 8

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Ring relations

There are 19 minimal relations of maximal degree 10:

  1. a_1_2·b_1_0
  2. b_1_32 + b_1_0·b_1_1
  3. b_1_0·b_1_12 + b_1_02·b_1_1
  4. b_1_12·b_1_3 + b_1_0·b_1_1·b_1_3 + a_1_2·b_1_1·b_1_3 + a_1_23
  5. b_1_0·a_3_10
  6. b_1_3·a_3_10 + a_1_2·b_3_11 + a_1_2·a_3_10
  7. a_1_23·b_1_12 + a_1_24·b_1_1 + a_1_25
  8. b_1_12·b_3_11 + b_1_0·b_1_1·b_3_11 + b_1_12·a_3_10 + a_1_2·b_1_1·b_3_11
       + a_1_2·b_1_1·a_3_10 + a_1_22·a_3_10
  9. a_3_10·b_3_11 + a_3_102 + a_1_23·a_3_10 + a_1_25·b_1_1 + c_4_16·a_1_2·b_1_3
  10. a_3_102 + a_1_2·b_1_12·a_3_10 + a_1_22·b_1_14 + a_1_22·b_1_1·a_3_10
       + a_1_25·b_1_1 + c_4_16·a_1_22
  11. b_3_112 + b_1_0·b_5_22 + b_1_02·b_1_3·b_3_11 + b_1_03·b_3_11 + a_3_102
       + a_1_23·b_3_11 + a_1_23·a_3_10 + c_4_16·b_1_02
  12. a_1_2·b_5_22 + a_1_2·b_1_12·a_3_10 + a_1_22·b_1_1·a_3_10 + a_1_23·b_3_11
       + c_4_16·a_1_2·b_1_1
  13. b_1_3·b_5_23 + b_1_1·b_5_22 + b_1_0·b_1_1·b_1_3·b_3_11 + b_1_02·b_1_1·b_3_11
       + b_1_04·b_1_1·b_1_3 + b_1_05·b_1_1 + b_1_13·a_3_10 + a_1_2·b_1_12·a_3_10
       + a_1_23·b_3_11 + a_1_23·a_3_10 + c_4_16·b_1_12 + c_4_16·b_1_0·b_1_3
  14. b_1_3·b_5_22 + b_1_0·b_5_23 + b_1_02·b_1_3·b_3_11 + b_1_02·b_1_1·b_3_11
       + b_1_04·b_1_1·b_1_3 + b_1_05·b_1_1 + a_1_23·a_3_10 + c_4_16·b_1_1·b_1_3
       + c_4_16·b_1_02
  15. b_1_12·b_5_22 + b_1_0·b_1_1·b_5_22 + b_1_14·a_3_10 + a_1_2·b_1_13·a_3_10
       + a_1_23·b_1_1·b_3_11 + c_4_16·b_1_13 + c_4_16·b_1_02·b_1_1
  16. a_3_10·b_5_22 + a_1_2·b_1_14·a_3_10 + a_1_22·b_1_16 + a_1_24·b_1_1·a_3_10
       + c_4_16·b_1_1·a_3_10 + c_4_16·a_1_22·b_1_12 + c_4_16·a_1_23·b_1_1
       + c_4_16·a_1_24
  17. b_5_232 + b_1_0·b_1_1·b_3_11·b_5_22 + b_1_04·b_1_1·b_5_22 + b_1_06·b_1_1·b_3_11
       + b_1_08·b_1_1·b_1_3 + b_1_17·a_3_10 + a_1_2·b_1_16·a_3_10 + a_1_22·b_1_18
       + a_1_22·b_1_15·a_3_10 + c_8_42·b_1_0·b_1_1 + c_4_16·b_1_16
       + c_4_16·b_1_0·b_1_1·b_1_3·b_3_11 + c_4_16·b_1_02·b_1_1·b_3_11
       + c_4_16·b_1_05·b_1_1 + c_4_16·a_1_22·b_1_14 + c_4_16·a_1_25·b_1_1
       + c_4_162·b_1_0·b_1_1 + c_4_162·b_1_02
  18. b_5_22·b_5_23 + b_1_0·b_1_1·b_3_11·b_5_22 + b_1_04·b_1_1·b_5_23
       + b_1_04·b_1_1·b_5_22 + b_1_05·b_5_23 + b_1_05·b_1_1·b_1_3·b_3_11 + b_1_09·b_1_1
       + b_1_12·a_3_10·b_5_23 + a_1_2·b_1_1·a_3_10·b_5_23 + c_8_42·b_1_0·b_1_3
       + c_4_16·b_1_1·b_5_23 + c_4_16·b_1_0·b_5_22 + c_4_16·b_1_0·b_1_1·b_1_3·b_3_11
       + c_4_16·b_1_02·b_1_3·b_3_11 + c_4_16·b_1_04·b_1_1·b_1_3 + c_4_16·b_1_06
       + c_4_16·a_1_25·b_1_1 + c_4_162·b_1_0·b_1_3 + c_4_162·b_1_0·b_1_1
  19. b_5_222 + b_1_02·b_3_11·b_5_22 + b_1_04·b_1_1·b_5_23 + b_1_05·b_5_23
       + b_1_05·b_5_22 + b_1_07·b_3_11 + b_1_08·b_1_1·b_1_3 + a_1_2·b_1_16·a_3_10
       + a_1_22·b_1_18 + a_1_22·b_1_15·a_3_10 + c_8_42·b_1_02
       + c_4_16·b_1_02·b_1_3·b_3_11 + c_4_16·b_1_03·b_3_11 + c_4_16·b_1_05·b_1_1
       + c_4_16·b_1_06 + c_4_16·a_1_22·b_1_14 + c_4_16·a_1_25·b_1_1 + c_4_162·b_1_12
       + c_4_162·b_1_02

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 11.
  • However, the last relation was already found in degree 10 and the last generator in degree 8.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_16, a Duflot regular element of degree 4
    2. c_8_42, a Duflot regular element of degree 8
    3. b_1_32 + b_1_12 + b_1_02, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 11].
  • 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 2

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_3_10 → 0, an element of degree 3
  6. b_3_11 → 0, an element of degree 3
  7. c_4_16 → c_1_04, an element of degree 4
  8. b_5_22 → 0, an element of degree 5
  9. b_5_23 → 0, an element of degree 5
  10. c_8_42 → c_1_18 + c_1_08, an element of degree 8

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_1_1 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_3_10 → 0, an element of degree 3
  6. b_3_11 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  8. b_5_22 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  9. b_5_23 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  10. c_8_42 → 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_16·c_1_22 + c_1_18 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22 + c_1_03·c_1_25
       + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_05·c_1_23
       + c_1_06·c_1_22 + c_1_08, an element of degree 8

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. a_3_10 → 0, an element of degree 3
  6. b_3_11 → 0, an element of degree 3
  7. c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  8. b_5_22 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  9. b_5_23 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  10. c_8_42 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_02·c_1_26 + c_1_04·c_1_24
       + c_1_08, an element of degree 8

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, an element of degree 1
  3. b_1_1 → c_1_2, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. a_3_10 → 0, an element of degree 3
  6. b_3_11 → c_1_23 + c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  8. b_5_22 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  9. b_5_23 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  10. c_8_42 → c_1_28 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22 + c_1_18
       + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24 + c_1_04·c_1_24
       + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + 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