Cohomology of group number 1437 of order 128

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General information on the group

  • The group has 4 minimal generators and exponent 4.
  • It is non-abelian.
  • It has p-Rank 4.
  • Its center has rank 3.
  • It has 3 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 3.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t6  −  t5  +  2·t4  +  2·t3  +  t2  +  2·t  +  1
    (t  +  1)2 · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-∞,-5,-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 4:

  1. a_1_0, a nilpotent 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. c_2_7, a Duflot regular element of degree 2
  6. b_3_10, an element of degree 3
  7. b_3_11, an element of degree 3
  8. b_3_12, an element of degree 3
  9. b_3_13, an element of degree 3
  10. c_4_22, a Duflot regular element of degree 4
  11. c_4_23, a Duflot regular element of degree 4

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Ring relations

There are 19 minimal relations of maximal degree 6:

  1. b_1_1·b_1_2
  2. b_1_32 + b_1_2·b_1_3 + a_1_0·b_1_2 + a_1_0·b_1_1
  3. b_1_1·b_1_3 + a_1_0·b_1_3 + a_1_0·b_1_2 + a_1_02
  4. a_1_02·b_1_2 + a_1_02·b_1_1 + a_1_03
  5. a_1_02·b_1_3
  6. b_1_3·b_3_12 + b_1_3·b_3_10 + b_1_2·b_3_12 + b_1_2·b_3_10 + a_1_0·b_3_12 + a_1_0·b_3_11
       + c_2_7·a_1_0·b_1_2
  7. b_1_1·b_3_12 + b_1_1·b_3_11 + a_1_0·b_3_11 + a_1_0·b_3_10 + c_2_7·a_1_0·b_1_2
  8. b_1_3·b_3_13 + b_1_3·b_3_11 + b_1_3·b_3_10 + b_1_2·b_3_12 + b_1_2·b_3_10 + a_1_0·b_3_10
       + c_2_7·b_1_2·b_1_3
  9. b_1_3·b_3_12 + b_1_2·b_3_13 + b_1_2·b_3_11 + b_1_23·b_1_3 + b_1_24 + a_1_0·b_3_11
       + a_1_0·b_3_10 + a_1_0·b_1_23 + c_2_7·b_1_22 + c_2_7·a_1_0·b_1_2
  10. b_1_3·b_3_12 + b_1_3·b_3_11 + b_1_2·b_3_12 + b_1_2·b_3_10 + a_1_0·b_3_13
       + c_2_7·b_1_2·b_1_3
  11. b_1_1·b_3_13 + b_1_1·b_3_11 + a_1_0·b_3_11 + a_1_0·b_3_10 + c_2_7·a_1_0·b_1_2
  12. a_1_02·b_3_10
  13. b_3_132 + b_3_122 + b_1_23·b_3_12 + b_1_23·b_3_11 + a_1_0·b_1_22·b_3_12
       + a_1_0·b_1_22·b_3_10 + c_4_22·b_1_22 + c_2_7·b_1_24 + c_2_7·a_1_0·b_1_23
       + c_2_72·b_1_22
  14. b_3_132 + b_3_122 + b_3_11·b_3_13 + b_3_11·b_3_12 + b_3_10·b_3_13 + b_3_10·b_3_12
       + b_1_23·b_3_12 + b_1_23·b_3_11 + b_1_25·b_1_3 + b_1_26 + a_1_0·b_1_22·b_3_12
       + a_1_0·b_1_22·b_3_10 + a_1_0·b_1_25 + c_4_22·b_1_2·b_1_3 + c_2_7·b_1_2·b_3_13
       + c_2_7·b_1_2·b_3_12 + c_2_7·b_1_24 + c_4_22·a_1_0·b_1_2 + c_2_7·a_1_0·b_1_23
       + c_2_72·b_1_2·b_1_3 + c_2_72·a_1_0·b_1_2
  15. b_3_132 + b_3_12·b_3_13 + b_3_11·b_3_13 + b_3_11·b_3_12 + b_3_112 + b_3_10·b_3_13
       + b_3_10·b_3_11 + b_1_23·b_3_13 + b_1_23·b_3_11 + b_1_23·b_3_10 + b_1_13·b_3_11
       + b_1_16 + a_1_0·b_1_22·b_3_12 + a_1_0·b_1_22·b_3_10 + c_4_23·b_1_2·b_1_3
       + c_4_23·b_1_22 + c_4_22·b_1_2·b_1_3 + c_4_22·b_1_12 + c_2_7·b_1_2·b_3_13
       + c_2_7·b_1_2·b_3_12 + c_2_7·b_1_2·b_3_10 + c_2_7·b_1_24 + c_2_7·b_1_14
       + c_4_23·a_1_0·b_1_3 + c_4_22·a_1_0·b_1_1 + c_2_7·a_1_0·b_1_23 + c_4_22·a_1_02
       + c_2_72·b_1_2·b_1_3 + c_2_72·b_1_22 + c_2_72·b_1_12 + c_2_72·a_1_0·b_1_1
       + c_2_72·a_1_02
  16. b_3_132 + b_3_12·b_3_13 + b_3_122 + b_3_11·b_3_13 + b_3_10·b_3_13 + b_3_10·b_3_12
       + b_1_23·b_3_11 + b_1_25·b_1_3 + b_1_26 + a_1_0·b_1_22·b_3_12 + a_1_0·b_1_22·b_3_10
       + a_1_0·b_1_25 + c_2_7·b_1_2·b_3_13 + c_2_7·b_1_24 + c_4_23·a_1_0·b_1_3
       + c_4_23·a_1_0·b_1_2 + c_4_22·a_1_0·b_1_3 + c_4_22·a_1_02 + c_2_72·a_1_0·b_1_3
       + c_2_72·a_1_02
  17. b_3_122 + b_3_11·b_3_13 + b_3_11·b_3_12 + b_3_112 + b_3_10·b_3_13 + b_3_10·b_3_12
       + b_1_23·b_3_13 + b_1_23·b_3_12 + b_1_25·b_1_3 + b_1_26 + a_1_0·b_1_22·b_3_12
       + a_1_0·b_1_22·b_3_10 + a_1_0·b_1_25 + c_2_7·b_1_2·b_3_13 + c_2_7·b_1_2·b_3_12
       + c_4_22·a_1_0·b_1_1 + c_4_23·a_1_02 + c_2_72·b_1_22 + c_2_72·a_1_0·b_1_1
  18. b_3_132 + b_3_122 + b_3_11·b_3_12 + b_3_112 + b_3_10·b_3_12 + b_3_10·b_3_11
       + b_1_23·b_3_12 + b_1_23·b_3_10 + b_1_25·b_1_3 + b_1_26 + a_1_0·b_1_22·b_3_12
       + a_1_0·b_1_22·b_3_10 + a_1_0·b_1_25 + c_4_22·b_1_2·b_1_3 + c_2_7·b_1_2·b_3_12
       + c_2_7·b_1_2·b_3_10 + c_4_23·a_1_0·b_1_1 + c_4_22·a_1_0·b_1_3 + c_4_22·a_1_0·b_1_1
       + c_2_7·a_1_0·b_1_23 + c_2_72·b_1_2·b_1_3 + c_2_72·b_1_22 + c_2_72·a_1_0·b_1_3
       + c_2_72·a_1_0·b_1_1
  19. b_3_11·b_3_13 + b_3_11·b_3_12 + b_3_112 + b_3_10·b_3_13 + b_3_10·b_3_12 + b_3_102
       + b_1_23·b_3_11 + b_1_23·b_3_10 + b_1_13·b_3_11 + b_1_16 + c_4_23·b_1_12
       + c_2_7·b_1_2·b_3_13 + c_2_7·b_1_2·b_3_12 + c_2_7·b_1_24 + c_4_22·a_1_0·b_1_1
       + c_4_22·a_1_02 + c_2_72·b_1_22 + c_2_72·a_1_0·b_1_1 + c_2_72·a_1_02

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 8.
  • However, the last relation was already found in degree 6 and the last generator in degree 4.
  • The following is a filter regular homogeneous system of parameters:
    1. c_2_7, a Duflot regular element of degree 2
    2. c_4_22, a Duflot regular element of degree 4
    3. c_4_23, a Duflot regular element of degree 4
    4. b_1_22 + b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 8].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is 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_1_2 → 0, an element of degree 1
  4. b_1_3 → 0, an element of degree 1
  5. c_2_7 → c_1_22, an element of degree 2
  6. b_3_10 → 0, an element of degree 3
  7. b_3_11 → 0, an element of degree 3
  8. b_3_12 → 0, an element of degree 3
  9. b_3_13 → 0, an element of degree 3
  10. c_4_22 → c_1_24 + c_1_14, an element of degree 4
  11. c_4_23 → c_1_04, an element of degree 4

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_3, 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. c_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_3_10 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  7. b_3_11 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  8. b_3_12 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  9. b_3_13 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  10. c_4_22 → c_1_34 + c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_14, an element of degree 4
  11. c_4_23 → c_1_34 + c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_02·c_1_32 + c_1_04, an element of degree 4

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 → 0, 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. c_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_3_10 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  7. b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  8. b_3_12 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  9. b_3_13 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  10. c_4_22 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_14, an element of degree 4
  11. c_4_23 → c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_02·c_1_32 + c_1_04, an element of degree 4

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 → 0, an element of degree 1
  3. b_1_2 → c_1_3, an element of degree 1
  4. b_1_3 → c_1_3, an element of degree 1
  5. c_2_7 → c_1_2·c_1_3 + c_1_22, an element of degree 2
  6. b_3_10 → c_1_12·c_1_3, an element of degree 3
  7. b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_12·c_1_3, an element of degree 3
  8. b_3_12 → c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
  9. b_3_13 → c_1_1·c_1_32, an element of degree 3
  10. c_4_22 → c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_14, an element of degree 4
  11. c_4_23 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_02·c_1_32
       + c_1_04, 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