Cohomology of group number 2191 of order 128

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General information on the group

  • The group has 5 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 2.
  • It has 4 conjugacy classes of maximal elementary abelian subgroups, which are all 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
    (2) · (t4  −  1/2·t3  +  1/2·t2  +  1/2·t  +  1/2)
    (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-4,-6,-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 10 minimal generators of maximal degree 4:

  1. b_1_0, an element of degree 1
  2. b_1_1, an element of degree 1
  3. b_1_2, an element of degree 1
  4. b_1_3, an element of degree 1
  5. b_1_4, an element of degree 1
  6. b_4_31, an element of degree 4
  7. b_4_32, an element of degree 4
  8. b_4_33, an element of degree 4
  9. c_4_34, a Duflot regular element of degree 4
  10. c_4_35, a Duflot regular element of degree 4

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Ring relations

There are 15 minimal relations of maximal degree 8:

  1. b_1_1·b_1_2 + b_1_0·b_1_3 + b_1_0·b_1_1
  2. b_1_42 + b_1_0·b_1_2
  3. b_1_0·b_1_22 + b_1_02·b_1_2
  4. b_1_0·b_1_32 + b_1_0·b_1_2·b_1_3 + b_1_0·b_1_1·b_1_3
  5. b_4_32·b_1_2
  6. b_1_03·b_1_1·b_1_3 + b_4_32·b_1_0 + b_4_31·b_1_2 + b_4_31·b_1_0
  7. b_1_02·b_1_12·b_1_3 + b_4_32·b_1_1 + b_4_31·b_1_3
  8. b_4_33·b_1_0 + b_4_31·b_1_2 + b_4_31·b_1_0
  9. b_1_02·b_1_12·b_1_3 + b_4_33·b_1_1 + b_4_32·b_1_3
  10. b_1_04·b_1_14 + b_1_06·b_1_12 + b_4_32·b_1_02·b_1_12 + b_4_32·b_1_04
       + b_4_31·b_1_02·b_1_1·b_1_3 + b_4_31·b_1_02·b_1_12 + b_4_31·b_1_03·b_1_2
       + b_4_31·b_1_03·b_1_1 + b_4_31·b_1_04 + b_4_312 + c_4_35·b_1_14
       + c_4_35·b_1_02·b_1_12 + c_4_34·b_1_04
  11. b_4_31·b_1_03·b_1_3 + b_4_31·b_4_32 + c_4_35·b_1_13·b_1_3
       + c_4_35·b_1_02·b_1_1·b_1_3 + c_4_34·b_1_03·b_1_2 + c_4_34·b_1_04
  12. b_1_04·b_1_14 + b_1_06·b_1_12 + b_4_32·b_1_04 + b_4_322
       + b_4_31·b_1_02·b_1_1·b_1_3 + b_4_31·b_1_02·b_1_12 + b_4_31·b_1_03·b_1_3
       + b_4_31·b_1_03·b_1_2 + b_4_31·b_1_03·b_1_1 + b_4_31·b_1_04 + b_4_312
       + c_4_35·b_1_12·b_1_32 + c_4_35·b_1_14 + c_4_35·b_1_02·b_1_1·b_1_3
       + c_4_35·b_1_02·b_1_12 + c_4_34·b_1_03·b_1_2
  13. b_4_31·b_1_02·b_1_1·b_1_3 + b_4_31·b_1_03·b_1_3 + b_4_31·b_4_33
       + c_4_35·b_1_12·b_1_32 + c_4_35·b_1_02·b_1_1·b_1_3 + c_4_34·b_1_03·b_1_2
       + c_4_34·b_1_04
  14. b_4_33·b_1_22·b_1_32 + b_4_33·b_1_23·b_1_3 + b_4_332
       + b_4_31·b_1_02·b_1_1·b_1_3 + b_4_31·b_1_03·b_1_3 + c_4_35·b_1_34
       + c_4_35·b_1_22·b_1_32 + c_4_35·b_1_02·b_1_1·b_1_3 + c_4_34·b_1_24
       + c_4_34·b_1_04
  15. b_4_32·b_4_33 + b_4_31·b_1_03·b_1_3 + c_4_35·b_1_1·b_1_33
       + c_4_35·b_1_02·b_1_1·b_1_3 + c_4_34·b_1_03·b_1_2 + c_4_34·b_1_04

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • 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_34, a Duflot regular element of degree 4
    2. c_4_35, a Duflot regular element of degree 4
    3. b_1_32 + b_1_2·b_1_4 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_3 + b_1_12 + b_1_0·b_1_1
         + b_1_02, an element of degree 2
    4. b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_1·b_1_32 + b_1_12·b_1_4 + b_1_12·b_1_3
         + b_1_0·b_1_2·b_1_3 + b_1_0·b_1_1·b_1_3 + b_1_02·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 4, 9].
  • 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 2 terms of the above HSOP, together with 2 elements of degree 2.

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

  1. b_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. b_1_3 → 0, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → 0, an element of degree 4
  7. b_4_32 → 0, an element of degree 4
  8. b_4_33 → 0, an element of degree 4
  9. c_4_34 → c_1_04, an element of degree 4
  10. c_4_35 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → c_1_2, an element of degree 1
  2. b_1_1 → c_1_3, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  7. b_4_32 → c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_23
       + c_1_02·c_1_22, an element of degree 4
  8. b_4_33 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  9. c_4_34 → 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
  10. c_4_35 → c_1_12·c_1_22 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → c_1_2, an element of degree 1
  3. b_1_2 → 0, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → c_1_12·c_1_22, an element of degree 4
  7. b_4_32 → c_1_12·c_1_2·c_1_3, an element of degree 4
  8. b_4_33 → c_1_12·c_1_32, an element of degree 4
  9. c_4_34 → 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
  10. c_4_35 → c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → c_1_3, 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. b_1_3 → 0, an element of degree 1
  5. b_1_4 → c_1_3, an element of degree 1
  6. b_4_31 → c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
  7. b_4_32 → 0, an element of degree 4
  8. b_4_33 → 0, an element of degree 4
  9. c_4_34 → 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
  10. c_4_35 → c_1_34 + c_1_12·c_1_32 + c_1_14, an element of degree 4

Restriction map to a maximal el. ab. subgp. of rank 4

  1. b_1_0 → 0, an element of degree 1
  2. b_1_1 → 0, an element of degree 1
  3. b_1_2 → c_1_2, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. b_1_4 → 0, an element of degree 1
  6. b_4_31 → 0, an element of degree 4
  7. b_4_32 → 0, an element of degree 4
  8. b_4_33 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  9. c_4_34 → 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
  10. c_4_35 → c_1_12·c_1_22 + c_1_14, an element of degree 4

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