Cohomology of group number 1786 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 4.
  • Its center has rank 2.
  • It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3 and 4, respectively.

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
    ( − 1) · (t6  −  t5  −  t4  +  t3  −  1)
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-6,-4,-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 8:

  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. c_2_8, a Duflot regular element of degree 2
  6. a_4_8, a nilpotent element of degree 4
  7. b_5_29, an element of degree 5
  8. b_5_30, an element of degree 5
  9. b_5_31, an element of degree 5
  10. a_8_41, a nilpotent element of degree 8
  11. c_8_77, a Duflot regular element of degree 8

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Ring relations

There are 29 minimal relations of maximal degree 16:

  1. b_1_12 + b_1_0·b_1_1
  2. b_1_0·b_1_2
  3. b_1_1·b_1_22 + b_1_0·b_1_32 + b_1_02·b_1_1
  4. b_1_0·b_1_34 + b_1_04·b_1_1
  5. a_4_8·b_1_0
  6. a_4_8·b_1_1
  7. b_1_2·b_5_29
  8. b_1_1·b_5_29 + b_1_0·b_5_30 + b_1_05·b_1_1
  9. b_1_1·b_5_30 + b_1_1·b_5_29 + b_1_05·b_1_1
  10. b_1_1·b_5_29 + b_1_0·b_5_31 + b_1_05·b_1_1
  11. b_1_2·b_5_30 + b_1_23·b_1_33 + b_1_1·b_5_31 + b_1_1·b_5_29 + b_1_05·b_1_1
       + a_4_8·b_1_32 + c_2_8·b_1_1·b_1_2·b_1_32
  12. b_1_32·b_5_29 + b_1_1·b_1_2·b_5_31 + b_1_02·b_5_30 + b_1_06·b_1_1
  13. a_4_82
  14. a_4_8·b_5_29
  15. a_4_8·b_5_30 + a_4_8·b_1_22·b_1_33
  16. b_1_1·b_1_33·b_5_31 + b_1_1·b_1_2·b_1_32·b_5_31 + b_1_03·b_1_3·b_5_30
       + a_8_41·b_1_2 + a_4_8·b_5_31 + a_4_8·b_1_35 + a_4_8·b_1_2·b_1_34
       + a_4_8·b_1_24·b_1_3 + c_2_8·b_1_1·b_1_3·b_5_31 + c_2_8·b_1_1·b_1_36
       + c_2_8·b_1_1·b_1_2·b_1_35 + c_2_8·b_1_0·b_1_3·b_5_30 + c_2_8·b_1_06·b_1_1
       + c_2_8·a_4_8·b_1_33 + c_2_8·a_4_8·b_1_22·b_1_3 + c_2_82·a_4_8·b_1_2
  17. a_8_41·b_1_0 + c_2_8·b_1_0·b_1_3·b_5_30 + c_2_8·b_1_0·b_1_3·b_5_29
       + c_2_8·b_1_05·b_1_1·b_1_3 + c_2_82·b_1_03·b_1_1·b_1_3 + c_2_82·b_1_04·b_1_1
  18. a_8_41·b_1_1 + c_2_82·b_1_03·b_1_1·b_1_3 + c_2_82·b_1_04·b_1_1
  19. b_5_302 + b_5_29·b_5_30 + b_1_24·b_1_36 + b_1_05·b_5_30
  20. b_5_302 + b_5_29·b_5_31 + b_1_24·b_1_36 + b_1_05·b_5_30
       + c_2_8·b_1_1·b_1_2·b_1_36
  21. b_5_292 + b_1_09·b_1_1 + c_8_77·b_1_02 + c_2_8·b_1_03·b_5_29
       + c_2_82·b_1_05·b_1_1 + c_2_83·b_1_03·b_1_1
  22. b_5_302 + b_1_24·b_1_36 + c_8_77·b_1_0·b_1_1 + c_2_8·b_1_03·b_5_30
       + c_2_8·b_1_07·b_1_1 + c_2_82·b_1_05·b_1_1 + c_2_83·b_1_03·b_1_1
  23. b_5_312 + b_5_302 + b_1_2·b_1_34·b_5_31 + b_1_23·b_1_32·b_5_31 + b_1_23·b_1_37
       + b_1_24·b_1_36 + b_1_25·b_5_31 + b_1_25·b_1_35 + b_1_26·b_1_34
       + b_1_27·b_1_33 + b_1_29·b_1_3 + a_8_41·b_1_2·b_1_3 + a_8_41·b_1_22
       + a_4_8·b_1_3·b_5_31 + a_4_8·b_1_2·b_5_31 + a_4_8·b_1_25·b_1_3 + a_4_8·b_1_26
       + c_8_77·b_1_22 + c_2_8·b_1_38 + c_2_8·b_1_22·b_1_36 + c_2_8·b_1_23·b_5_31
       + c_2_8·b_1_24·b_1_34 + c_2_8·b_1_25·b_1_33 + c_2_8·b_1_26·b_1_32
       + c_2_8·b_1_1·b_1_32·b_5_31 + c_2_8·b_1_1·b_1_37 + c_2_8·b_1_1·b_1_2·b_1_3·b_5_31
       + c_2_8·b_1_1·b_1_2·b_1_36 + c_2_8·b_1_03·b_5_30 + c_2_8·b_1_06·b_1_1·b_1_3
       + c_2_8·b_1_07·b_1_1 + c_2_8·a_4_8·b_1_34 + c_2_8·a_4_8·b_1_2·b_1_33
       + c_2_8·a_4_8·b_1_23·b_1_3 + c_2_82·a_4_8·b_1_2·b_1_3 + c_2_82·a_4_8·b_1_22
       + c_2_83·b_1_22·b_1_32
  24. b_5_30·b_5_31 + b_5_302 + b_1_35·b_5_30 + b_1_22·b_1_33·b_5_31 + b_1_22·b_1_38
       + b_1_24·b_1_36 + b_1_04·b_1_3·b_5_30 + a_8_41·b_1_32 + a_4_8·b_1_36
       + a_4_8·b_1_23·b_1_33 + c_8_77·b_1_1·b_1_2 + c_2_8·b_1_33·b_5_30
       + c_2_8·b_1_22·b_1_36 + c_2_8·b_1_1·b_1_32·b_5_31
       + c_2_8·b_1_1·b_1_2·b_1_3·b_5_31 + c_2_8·b_1_1·b_1_2·b_1_36
       + c_2_8·b_1_02·b_1_3·b_5_30 + c_2_8·b_1_03·b_5_30 + c_2_8·a_4_8·b_1_2·b_1_33
       + c_2_82·b_1_1·b_1_35 + c_2_82·b_1_05·b_1_1 + c_2_82·a_4_8·b_1_32
       + c_2_83·b_1_1·b_1_2·b_1_32
  25. a_4_8·a_8_41
  26. a_8_41·b_5_30 + a_8_41·b_1_22·b_1_33 + c_2_82·b_1_03·b_1_3·b_5_30
       + c_2_82·b_1_04·b_5_30
  27. a_8_41·b_5_29 + c_2_8·c_8_77·b_1_0·b_1_1·b_1_3 + c_2_8·c_8_77·b_1_02·b_1_3
       + c_2_82·b_1_03·b_1_3·b_5_29 + c_2_82·b_1_04·b_5_30 + c_2_82·b_1_08·b_1_1
  28. b_1_38·b_5_30 + b_1_22·b_1_311 + b_1_08·b_5_30 + a_8_41·b_5_31 + a_8_41·b_5_30
       + a_8_41·b_1_35 + a_4_8·b_1_39 + a_4_8·b_1_2·b_1_33·b_5_31 + a_4_8·b_1_2·b_1_38
       + a_4_8·b_1_22·b_1_32·b_5_31 + a_4_8·b_1_23·b_1_3·b_5_31 + a_4_8·b_1_23·b_1_36
       + a_4_8·b_1_24·b_5_31 + a_4_8·b_1_24·b_1_35 + a_4_8·b_1_26·b_1_33
       + a_4_8·b_1_28·b_1_3 + c_8_77·b_1_1·b_1_2·b_1_33 + c_2_8·b_1_36·b_5_30
       + c_2_8·b_1_22·b_1_39 + c_2_8·b_1_1·b_1_310 + c_2_8·b_1_06·b_5_30
       + c_2_8·b_1_010·b_1_1 + a_4_8·c_8_77·b_1_2 + c_2_8·a_8_41·b_1_33
       + c_2_8·a_8_41·b_1_2·b_1_32 + c_2_8·a_8_41·b_1_22·b_1_3
       + c_2_8·a_4_8·b_1_32·b_5_31 + c_2_8·a_4_8·b_1_2·b_1_36
       + c_2_8·a_4_8·b_1_22·b_5_31 + c_2_8·a_4_8·b_1_22·b_1_35
       + c_2_8·a_4_8·b_1_23·b_1_34 + c_2_8·c_8_77·b_1_1·b_1_2·b_1_3
       + c_2_82·b_1_34·b_5_30 + c_2_82·b_1_22·b_1_37 + c_2_82·b_1_1·b_1_38
       + c_2_82·b_1_1·b_1_2·b_1_32·b_5_31 + c_2_82·b_1_04·b_5_30
       + c_2_82·b_1_07·b_1_1·b_1_3 + c_2_82·a_8_41·b_1_2 + c_2_82·a_4_8·b_1_35
       + c_2_82·a_4_8·b_1_2·b_1_34 + c_2_82·a_4_8·b_1_23·b_1_32
       + c_2_82·a_4_8·b_1_24·b_1_3 + c_2_83·b_1_1·b_1_3·b_5_31
       + c_2_83·b_1_0·b_1_3·b_5_30 + c_2_83·b_1_05·b_1_1·b_1_3 + c_2_83·b_1_06·b_1_1
       + c_2_84·b_1_1·b_1_2·b_1_33 + c_2_84·a_4_8·b_1_2
  29. a_8_412

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Data used for Benson′s test

  • Benson′s completion test succeeded in degree 16.
  • 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_2_8, a Duflot regular element of degree 2
    2. c_8_77, a Duflot regular element of degree 8
    3. b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_3 + b_1_02, an element of degree 2
    4. b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_0·b_1_32 + b_1_02·b_1_1, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 11].
  • 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 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. c_2_8 → c_1_02, an element of degree 2
  6. a_4_8 → 0, an element of degree 4
  7. b_5_29 → 0, an element of degree 5
  8. b_5_30 → 0, an element of degree 5
  9. b_5_31 → 0, an element of degree 5
  10. a_8_41 → 0, an element of degree 8
  11. c_8_77 → c_1_18, an element of degree 8

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

  1. b_1_0 → c_1_2, 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_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  6. a_4_8 → 0, an element of degree 4
  7. b_5_29 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  8. b_5_30 → 0, an element of degree 5
  9. b_5_31 → 0, an element of degree 5
  10. a_8_41 → 0, an element of degree 8
  11. c_8_77 → c_1_14·c_1_24 + 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_12·c_1_24 + c_1_02·c_1_14·c_1_22, an element of degree 8

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

  1. b_1_0 → c_1_2, 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_2, an element of degree 1
  5. c_2_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  6. a_4_8 → 0, an element of degree 4
  7. b_5_29 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  8. b_5_30 → c_1_12·c_1_23 + 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_31 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  10. a_8_41 → 0, an element of degree 8
  11. c_8_77 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_0·c_1_12·c_1_25
       + c_1_0·c_1_14·c_1_23 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
       + c_1_03·c_1_25 + c_1_05·c_1_23 + c_1_06·c_1_22, an element of degree 8

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. c_2_8 → c_1_0·c_1_2 + c_1_02, an element of degree 2
  6. a_4_8 → 0, an element of degree 4
  7. b_5_29 → 0, an element of degree 5
  8. b_5_30 → c_1_22·c_1_33, an element of degree 5
  9. b_5_31 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_12·c_1_23 + c_1_14·c_1_2
       + c_1_0·c_1_34 + c_1_02·c_1_2·c_1_32, an element of degree 5
  10. a_8_41 → 0, an element of degree 8
  11. c_8_77 → c_1_23·c_1_35 + c_1_25·c_1_33 + c_1_26·c_1_32 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_24·c_1_32 + c_1_12·c_1_26 + c_1_14·c_1_34
       + c_1_14·c_1_22·c_1_32 + c_1_18 + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_25·c_1_32
       + c_1_0·c_1_26·c_1_3 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
       + c_1_02·c_1_25·c_1_3 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
       + c_1_03·c_1_2·c_1_34 + c_1_04·c_1_34 + c_1_05·c_1_2·c_1_32 + c_1_06·c_1_32, an element of degree 8

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