David J. Green

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Cohomology of group number 1469 of order 128

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

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 2 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

  ( − 1) · (t3  +  t  +  1) · (t4  −  3·t3  −  t  −  1)

  (t  +  1)2 · (t  −  1)4 · (t2  +  1)3

• The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring generators

The cohomology ring has 16 minimal generators of maximal degree 5:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_1_2, an element of degree 1
4. b_1_3, an element of degree 1
5. a_3_1, a nilpotent element of degree 3
6. b_3_5, an element of degree 3
7. b_3_6, an element of degree 3
8. b_3_8, an element of degree 3
9. b_3_9, an element of degree 3
10. b_3_10, an element of degree 3
11. b_3_11, an element of degree 3
12. c_4_19, a Duflot regular element of degree 4
13. c_4_20, a Duflot regular element of degree 4
14. c_4_21, a Duflot regular element of degree 4
15. b_5_31, an element of degree 5
16. b_5_32, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 62 minimal relations of maximal degree 10:

1. a_1_1·b_1_2 + a_1_1² + a_1_0²
2. b_1_2·b_1_3 + a_1_1·b_1_3 + a_1_1² + a_1_0·a_1_1 + a_1_0²
3. a_1_0·b_1_3 + a_1_0·b_1_2 + a_1_1² + a_1_0·a_1_1
4. a_1_1³
5. a_1_0·a_1_1² + a_1_0²·a_1_1
6. a_1_1²·b_1_3
7. b_1_3·b_3_5 + a_1_1·b_3_6 + a_1_0·b_3_5
8. b_1_3·b_3_5 + b_1_2·b_3_6 + b_1_2⁴ + a_1_0·b_3_6
9. b_1_3·b_3_5 + b_1_2·b_3_8 + b_1_2·b_3_6 + b_1_2·b_3_5 + a_1_1·b_3_8 + a_1_0·b_3_8
10. a_1_1·b_3_9 + a_1_1·b_3_8 + a_1_1·a_3_1 + a_1_0·a_3_1
11. b_1_3·b_3_5 + b_1_2·b_3_8 + b_1_2·b_3_6 + b_1_2·b_3_5 + b_1_2·a_3_1 + a_1_1·b_3_8
       + a_1_0·b_3_9 + a_1_1·a_3_1 + a_1_0·a_3_1
12. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_2·b_3_6 + b_1_2⁴ + b_1_2·a_3_1 + a_1_1·b_3_8
       + a_1_0·b_3_5 + a_1_0·a_3_1
13. b_1_2·b_3_8 + b_1_2·b_3_5 + b_1_2⁴ + a_1_1·b_3_10 + a_1_1·a_3_1
14. b_1_2·b_3_8 + b_1_2·b_3_5 + b_1_2⁴ + a_1_1·b_3_8 + a_1_1·b_3_5 + a_1_0·b_3_10
       + a_1_0·a_3_1
15. b_1_3·b_3_10 + b_1_3·b_3_8 + b_1_2·b_3_6 + b_1_2⁴ + b_1_3·a_3_1 + a_1_1·b_3_8
       + a_1_0·b_3_5
16. b_1_3·b_3_5 + b_1_2·b_3_11 + b_1_2·b_3_10 + b_1_2·b_3_9 + b_1_2·b_3_8 + b_1_2·b_3_6
       + b_1_2·b_3_5 + b_1_3·a_3_1 + a_1_1·b_3_5 + a_1_0·b_3_5 + a_1_1·a_3_1 + a_1_0·a_3_1
17. b_1_3·a_3_1 + b_1_2·a_3_1 + a_1_1·b_3_11 + a_1_1·b_3_8 + a_1_0·a_3_1
18. b_1_3·b_3_5 + b_1_2·b_3_8 + b_1_2·b_3_6 + b_1_2·b_3_5 + b_1_2·a_3_1 + a_1_1·b_3_8
       + a_1_0·b_3_11 + a_1_0·a_3_1
19. a_1_1²·b_3_8 + a_1_1²·b_3_6 + a_1_1²·a_3_1 + a_1_0·a_1_1·a_3_1
20. a_1_1²·b_3_6 + a_1_0²·b_3_9 + a_1_0²·b_3_8 + a_1_0²·b_3_6 + a_1_0·a_1_1·a_3_1
       + a_1_0²·a_3_1
21. a_1_0²·b_3_10 + a_1_0²·b_3_8 + a_1_0²·b_3_6 + a_1_0²·b_3_5 + a_1_0²·a_3_1
22. b_3_6·b_3_10 + b_3_6·b_3_8 + b_3_5·b_3_8 + b_3_5·b_3_6 + b_3_5² + b_1_2³·b_3_10
       + b_1_2³·b_3_5 + b_1_2⁶ + a_3_1·b_3_6
23. b_3_8·b_3_10 + b_3_8·b_3_9 + b_3_5·b_3_10 + b_3_5·b_3_9 + b_3_5·b_3_8 + b_3_5²
       + b_1_2³·b_3_10 + b_1_2³·b_3_9 + b_1_2³·b_3_5 + a_3_1·b_3_10 + a_3_1²
24. b_3_9·b_3_11 + b_3_9·b_3_10 + b_3_9² + b_3_8·b_3_11 + b_3_5·b_3_10 + b_3_5·b_3_9
       + b_3_5·b_3_8 + b_3_5² + b_1_2³·b_3_10 + b_1_2³·b_3_9 + b_1_2³·b_3_5 + a_3_1·b_3_11
       + a_3_1·b_3_10 + a_3_1·b_3_5 + a_1_1·b_1_3²·b_3_6
25. b_3_10·b_3_11 + b_3_10² + b_3_9·b_3_11 + b_3_9² + b_3_8·b_3_9 + b_3_8² + b_3_5·b_3_11
       + b_3_5·b_3_10 + b_3_5·b_3_8 + b_1_2³·b_3_9 + b_1_2³·b_3_5 + b_1_2⁶ + a_3_1·b_3_9
       + a_3_1·b_3_5 + a_1_1·b_1_3²·b_3_6
26. b_3_10² + b_3_8² + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_8 + b_1_2³·b_3_9
       + b_1_2³·b_3_5 + a_3_1·b_3_5 + a_3_1² + c_4_19·b_1_2² + c_4_19·a_1_0·b_1_2
27. b_3_10² + b_3_8² + b_3_5·b_3_11 + b_3_5·b_3_10 + b_3_5·b_3_9 + b_3_5·b_3_8 + b_1_2⁶
       + a_3_1·b_3_10 + a_3_1·b_3_8 + a_3_1·b_3_6 + a_3_1·b_3_5 + c_4_19·b_1_2²
       + c_4_19·a_1_1² + c_4_19·a_1_0·a_1_1
28. b_3_10² + b_3_8² + b_3_5² + b_1_2³·b_3_5 + b_1_2⁶ + a_3_1² + c_4_19·b_1_2²
       + c_4_19·a_1_0²
29. b_3_8·b_3_9 + b_3_8² + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_9 + b_3_5² + b_1_2³·b_3_5
       + a_3_1·b_3_10 + a_3_1·b_3_8 + a_1_1·b_1_3²·b_3_8 + a_3_1² + c_4_19·a_1_1·b_1_3
30. b_3_8² + b_3_5² + b_1_3³·b_3_8 + b_1_2⁶ + a_1_1·b_1_3²·b_3_11
       + a_1_1·b_1_3²·b_3_6 + c_4_19·b_1_3²
31. b_3_11² + b_3_10² + b_3_8² + b_1_3³·b_3_6 + b_1_2³·b_3_10 + b_1_2³·b_3_9 + b_1_2⁶
       + a_1_1·b_1_3²·b_3_11 + c_4_20·b_1_2² + c_4_19·b_1_2² + c_4_19·a_1_1²
32. b_3_10² + b_3_8·b_3_9 + b_3_5·b_3_11 + b_3_5·b_3_10 + b_1_2³·b_3_9 + b_1_2³·b_3_5
       + a_3_1·b_3_11 + a_3_1·b_3_8 + a_3_1·b_3_6 + a_1_1·b_1_3²·b_3_6 + c_4_19·b_1_2²
       + c_4_20·a_1_0·b_1_2 + c_4_19·a_1_1²
33. b_3_11² + b_3_9² + b_3_8² + b_3_5² + b_1_3³·b_3_6 + b_1_2³·b_3_5 + b_1_2⁶
       + a_1_1·b_1_3²·b_3_11 + c_4_19·b_1_2² + c_4_20·a_1_1²
34. b_3_8·b_3_9 + b_3_8² + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_11 + b_3_5·b_3_10
       + b_3_5·b_3_8 + a_3_1·b_3_11 + a_3_1·b_3_9 + a_3_1·b_3_6 + a_3_1·b_3_5
       + a_1_1·b_1_3²·b_3_6 + c_4_20·a_1_0·a_1_1 + c_4_19·a_1_1²
35. b_3_10² + b_3_8² + b_3_5² + b_1_2³·b_3_5 + b_1_2⁶ + c_4_19·b_1_2²
       + c_4_20·a_1_0² + c_4_19·a_1_1²
36. b_3_5² + b_1_2³·b_3_10 + b_1_2³·b_3_9 + b_1_2³·b_3_5 + c_4_21·b_1_2²
       + c_4_21·a_1_0² + c_4_19·a_1_1²
37. b_3_10² + b_3_8·b_3_9 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_11 + b_3_5·b_3_10
       + b_3_5·b_3_8 + b_3_5·b_3_6 + b_1_2³·b_3_10 + b_1_2³·b_3_9 + b_1_2³·b_3_5 + b_1_2⁶
       + a_3_1·b_3_6 + a_3_1·b_3_5 + a_1_1·b_1_3²·b_3_11 + a_1_1·b_1_3²·b_3_6 + a_3_1²
       + c_4_21·b_1_2² + c_4_19·b_1_2² + c_4_21·a_1_1·b_1_3 + c_4_21·a_1_0·b_1_2
38. b_3_10² + b_3_6² + b_1_3³·b_3_11 + b_1_3³·b_3_6 + b_1_2³·b_3_5 + b_1_2⁶
       + a_1_1·b_1_3²·b_3_11 + a_3_1² + c_4_21·b_1_3² + c_4_19·b_1_2²
39. b_3_8·b_3_9 + b_3_8² + b_3_5·b_3_10 + b_3_5·b_3_8 + b_1_2·b_5_31 + b_1_2³·b_3_10
       + a_3_1·b_3_11 + a_3_1·b_3_8 + a_3_1·b_3_6 + a_3_1·b_3_5 + a_1_1·b_1_3²·b_3_11 + a_3_1²
       + c_4_19·b_1_2² + c_4_21·a_1_1²
40. b_3_8·b_3_9 + b_3_8² + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_11 + b_3_5·b_3_10
       + b_3_5·b_3_8 + b_3_5² + b_1_2³·b_3_10 + b_1_2³·b_3_9 + b_1_2³·b_3_5 + a_3_1·b_3_11
       + a_3_1·b_3_10 + a_3_1·b_3_9 + a_3_1·b_3_8 + a_3_1·b_3_5 + a_1_1·b_5_31
       + a_1_1·b_1_3²·b_3_11 + a_3_1² + c_4_21·b_1_2²
41. b_3_10² + b_3_8² + b_3_5² + b_1_2³·b_3_5 + b_1_2⁶ + a_3_1·b_3_5 + a_1_0·b_5_31
       + c_4_19·b_1_2² + c_4_21·a_1_0·b_1_2 + c_4_21·a_1_0·a_1_1 + c_4_19·a_1_1²
42. b_3_11² + b_3_10² + b_3_9² + b_3_8·b_3_9 + b_3_8² + b_3_6·b_3_11 + b_3_6·b_3_9
       + b_3_5·b_3_9 + b_1_3·b_5_31 + b_1_3³·b_3_11 + b_1_2³·b_3_5 + b_1_2⁶ + a_3_1·b_3_10
       + a_3_1·b_3_9 + a_1_1·b_1_3²·b_3_11 + a_1_1·b_1_3²·b_3_8 + a_3_1² + c_4_21·b_1_2²
       + c_4_21·a_1_1² + c_4_21·a_1_0·a_1_1 + c_4_19·a_1_1²
43. b_3_10² + b_3_9·b_3_10 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_11 + b_3_5·b_3_10
       + b_1_2·b_5_32 + b_1_2³·b_3_10 + b_1_2³·b_3_9 + a_3_1·b_3_11 + a_3_1·b_3_10 + a_3_1·b_3_5
       + a_1_1·b_1_3²·b_3_8 + c_4_21·b_1_2² + c_4_21·a_1_0·b_1_2 + c_4_20·a_1_1·b_1_3
       + c_4_21·a_1_1²
44. b_3_11² + b_3_9² + b_3_8·b_3_9 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_11 + b_3_5·b_3_10
       + b_3_5·b_3_8 + b_3_5² + b_1_3³·b_3_6 + b_1_2³·b_3_5 + b_1_2⁶ + a_3_1·b_3_8
       + a_3_1·b_3_5 + a_1_1·b_5_32 + a_1_1·b_1_3²·b_3_11 + a_1_1·b_1_3²·b_3_8
       + a_1_1·b_1_3²·b_3_6 + c_4_19·b_1_2² + c_4_20·a_1_1·b_1_3 + c_4_21·a_1_1²
       + c_4_21·a_1_0·a_1_1
45. b_3_11² + b_3_10² + b_3_9² + b_3_8·b_3_9 + b_3_8² + b_3_5·b_3_9 + b_3_5·b_3_8
       + b_3_5² + b_1_3³·b_3_6 + b_1_2³·b_3_10 + b_1_2⁶ + a_3_1·b_3_11 + a_3_1·b_3_8
       + a_3_1·b_3_5 + a_1_1·b_1_3²·b_3_11 + a_1_1·b_1_3²·b_3_6 + a_1_0·b_5_32
       + c_4_21·b_1_2² + c_4_21·a_1_0·a_1_1
46. b_3_10² + b_3_9·b_3_11 + b_3_9·b_3_10 + b_3_9² + b_3_8·b_3_9 + b_3_6·b_3_11
       + b_3_6·b_3_9 + b_3_5·b_3_9 + b_3_5·b_3_8 + b_3_5·b_3_6 + b_1_3·b_5_32 + b_1_3³·b_3_8
       + b_1_3³·b_3_6 + a_3_1·b_3_11 + a_3_1·b_3_10 + a_3_1·b_3_6 + a_1_1·b_1_3²·b_3_11
       + a_3_1² + c_4_21·b_1_2² + c_4_20·b_1_3² + c_4_19·b_1_2² + c_4_20·a_1_1·b_1_3
       + c_4_21·a_1_1² + c_4_21·a_1_0·a_1_1
47. a_1_1²·b_5_31 + a_1_0·a_1_1·b_5_32 + a_1_0·a_1_1·b_5_31 + a_1_0²·b_5_32
       + a_1_0²·b_5_31 + c_4_21·a_1_0²·a_1_1 + c_4_19·a_1_0²·a_1_1
48. b_3_5·b_5_31 + b_1_2⁵·b_3_10 + b_1_2⁵·b_3_9 + a_1_1·b_1_3·b_3_6·b_3_8
       + a_1_1·b_1_3⁴·b_3_11 + a_1_0³·b_5_31 + c_4_21·b_1_2·b_3_10 + c_4_21·b_1_2·b_3_9
       + c_4_20·b_1_2⁴ + c_4_19·b_1_2·b_3_5 + c_4_21·a_1_1·b_3_11 + c_4_21·a_1_1·b_3_8
       + c_4_21·a_1_1·b_1_3³ + c_4_21·a_1_0·b_3_6 + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_0·b_3_5
       + c_4_19·a_1_1·b_3_11 + c_4_19·a_1_1·b_3_8 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_5
       + c_4_21·a_1_1·a_3_1 + c_4_21·a_1_0·a_3_1 + c_4_19·a_1_1·a_3_1
49. b_3_10·b_5_31 + b_3_9·b_5_31 + b_1_2³·b_5_31 + b_1_2⁵·b_3_5 + b_1_2⁸ + a_3_1·b_5_31
       + a_1_1·b_1_3·b_3_6·b_3_8 + a_1_1·b_1_3⁴·b_3_11 + c_4_21·b_1_2·b_3_5 + c_4_21·b_1_2⁴
       + c_4_20·b_1_2·b_3_5 + c_4_20·b_1_2⁴ + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_2·b_3_9
       + c_4_19·b_1_2⁴ + c_4_21·a_1_1·b_3_11 + c_4_21·a_1_1·b_3_8 + c_4_21·a_1_1·b_1_3³
       + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_6
       + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_6
       + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_11 + c_4_19·a_1_1·b_3_8 + c_4_19·a_1_0·b_3_8
       + c_4_19·a_1_0·b_3_6 + c_4_21·a_1_1·a_3_1
50. b_3_6·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_3⁵·b_3_11 + b_1_2³·b_5_31
       + a_1_1·b_1_3²·b_5_31 + a_1_1·b_1_3⁴·b_3_11 + a_1_1·b_1_3⁴·b_3_6 + a_1_0³·b_5_31
       + c_4_21·b_1_3·b_3_11 + c_4_21·b_1_3·b_3_8 + c_4_21·b_1_3⁴ + c_4_19·b_1_3·b_3_11
       + c_4_19·b_1_3·b_3_8 + c_4_21·a_1_0·b_3_6 + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_0·b_3_6
       + c_4_19·a_1_1·b_3_6 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·a_3_1
51. b_3_6·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_3⁵·b_3_11 + b_1_2³·b_5_31 + a_3_1·b_5_32
       + a_1_1·b_1_3⁴·b_3_11 + a_1_1·b_1_3⁴·b_3_6 + a_1_0³·b_5_31 + c_4_21·b_1_3·b_3_11
       + c_4_21·b_1_3·b_3_8 + c_4_21·b_1_3⁴ + c_4_19·b_1_3·b_3_11 + c_4_19·b_1_3·b_3_8
       + c_4_21·a_1_1·b_1_3³ + c_4_21·a_1_0·b_3_6 + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_1·b_3_11
       + c_4_20·a_1_1·b_3_8 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_6
       + c_4_19·a_1_1·b_3_11 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8 + c_4_21·a_1_1·a_3_1
       + c_4_21·a_1_0·a_3_1 + c_4_20·a_1_0·a_3_1
52. b_3_11·b_5_32 + b_3_9·b_5_31 + b_1_3³·b_5_31 + b_1_3⁵·b_3_8 + b_1_2²·b_3_5·b_3_9
       + b_1_2³·b_5_31 + b_1_2⁵·b_3_9 + b_1_2⁵·b_3_5 + b_1_2⁸ + a_1_1·b_1_3⁴·b_3_6
       + a_1_0³·b_5_31 + c_4_21·b_1_3⁴ + c_4_21·b_1_2·b_3_9 + c_4_21·b_1_2⁴
       + c_4_20·b_1_3·b_3_11 + c_4_20·b_1_2·b_3_10 + c_4_20·b_1_2⁴ + c_4_19·b_1_3·b_3_6
       + c_4_19·b_1_3⁴ + c_4_19·b_1_2·b_3_9 + c_4_19·b_1_2⁴ + c_4_21·a_1_1·b_3_11
       + c_4_21·a_1_1·b_3_8 + c_4_21·a_1_0·b_3_9 + c_4_20·a_1_1·b_3_11 + c_4_20·a_1_1·b_3_8
       + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_6 + c_4_19·a_1_1·b_1_3³ + c_4_19·a_1_0·b_3_9
       + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_0·a_3_1 + c_4_20·a_1_1·a_3_1
       + c_4_20·a_1_0·a_3_1
53. b_3_9·b_5_31 + b_3_8·b_5_31 + b_3_5·b_5_32 + b_1_2²·b_3_5·b_3_9 + b_1_2⁵·b_3_10
       + b_1_2⁵·b_3_5 + b_1_2⁸ + a_3_1·b_5_31 + a_1_1·b_1_3⁴·b_3_8 + a_1_0³·b_5_31
       + c_4_21·b_1_2·b_3_10 + c_4_21·b_1_2⁴ + c_4_20·b_1_2·b_3_5 + c_4_20·b_1_2⁴
       + c_4_19·b_1_2·b_3_9 + c_4_19·b_1_2·b_3_5 + c_4_19·b_1_2⁴ + c_4_21·a_1_1·b_3_11
       + c_4_21·a_1_0·b_3_9 + c_4_20·a_1_1·b_3_6 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8
       + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_11 + c_4_19·a_1_1·b_1_3³
       + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6
       + c_4_21·a_1_0·a_3_1 + c_4_20·a_1_0·a_3_1 + c_4_19·a_1_1·a_3_1
54. b_3_11·b_5_31 + b_3_6·b_5_32 + b_3_6·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_3⁵·b_3_11
       + b_1_3⁵·b_3_8 + b_1_2³·b_5_32 + b_1_2⁵·b_3_5 + b_1_2⁸ + a_1_1·b_1_3·b_3_6·b_3_8
       + a_1_1·b_1_3⁴·b_3_11 + a_1_1·b_1_3⁴·b_3_8 + a_1_0³·b_5_31 + c_4_21·b_1_3⁴
       + c_4_21·b_1_2·b_3_5 + c_4_21·b_1_2⁴ + c_4_20·b_1_3·b_3_6 + c_4_20·b_1_2·b_3_5
       + c_4_20·b_1_2⁴ + c_4_19·b_1_3⁴ + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_2·b_3_9
       + c_4_19·b_1_2⁴ + c_4_21·a_1_1·b_3_11 + c_4_21·a_1_1·b_3_8 + c_4_21·a_1_1·b_3_6
       + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_6 + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_1·b_3_6
       + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_6 + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_8
       + c_4_19·a_1_1·b_3_6 + c_4_19·a_1_1·b_1_3³ + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6
       + c_4_21·a_1_1·a_3_1 + c_4_20·a_1_1·a_3_1
55. b_3_9·b_5_32 + b_3_8·b_5_31 + b_1_3⁵·b_3_8 + b_1_2³·b_5_32 + a_3_1·b_5_31
       + a_1_1·b_1_3·b_3_6·b_3_8 + c_4_21·b_1_2·b_3_5 + c_4_20·b_1_3·b_3_8
       + c_4_20·b_1_2·b_3_10 + c_4_20·b_1_2·b_3_5 + c_4_20·b_1_2⁴ + c_4_19·b_1_3·b_3_11
       + c_4_19·b_1_3⁴ + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_2·b_3_9 + c_4_21·a_1_1·b_3_8
       + c_4_21·a_1_1·b_1_3³ + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_1·b_3_8 + c_4_20·a_1_0·b_3_10
       + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_1·b_3_11 + c_4_19·a_1_1·b_3_8 + c_4_19·a_1_1·b_3_6
       + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_9 + c_4_21·a_1_1·a_3_1 + c_4_20·a_1_1·a_3_1
56. b_3_9·b_5_31 + b_3_8·b_5_32 + b_1_3⁵·b_3_8 + b_1_2²·b_3_5·b_3_9 + b_1_2³·b_5_32
       + b_1_2³·b_5_31 + b_1_2⁵·b_3_9 + b_1_2⁵·b_3_5 + b_1_2⁸ + a_3_1·b_5_31
       + a_1_1·b_1_3·b_3_6·b_3_8 + a_1_1·b_1_3⁴·b_3_8 + a_1_0³·b_5_31 + c_4_21·b_1_2·b_3_9
       + c_4_21·b_1_2⁴ + c_4_20·b_1_3·b_3_8 + c_4_20·b_1_2·b_3_5 + c_4_19·b_1_3·b_3_11
       + c_4_19·b_1_3⁴ + c_4_19·b_1_2·b_3_9 + c_4_19·b_1_2⁴ + c_4_21·a_1_1·b_1_3³
       + c_4_21·a_1_0·b_3_9 + c_4_20·a_1_1·b_3_8 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_9
       + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_1·b_3_6 + c_4_19·a_1_1·b_1_3³
       + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_1·a_3_1
57. b_3_10·b_5_32 + b_3_9·b_5_31 + b_3_6·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_3⁵·b_3_11
       + b_1_3⁵·b_3_8 + b_1_2²·b_3_5·b_3_9 + b_1_2³·b_5_32 + b_1_2³·b_5_31 + b_1_2⁵·b_3_10
       + b_1_2⁵·b_3_5 + b_1_2⁸ + a_1_1·b_1_3·b_3_6·b_3_8 + a_1_1·b_1_3⁴·b_3_11
       + a_1_1·b_1_3⁴·b_3_8 + a_1_1·b_1_3⁴·b_3_6 + a_1_0³·b_5_31 + c_4_21·b_1_3·b_3_11
       + c_4_21·b_1_3·b_3_8 + c_4_21·b_1_3⁴ + c_4_21·b_1_2·b_3_10 + c_4_21·b_1_2·b_3_5
       + c_4_21·b_1_2⁴ + c_4_20·b_1_3·b_3_8 + c_4_20·b_1_2·b_3_5 + c_4_20·b_1_2⁴
       + c_4_19·b_1_3·b_3_8 + c_4_19·b_1_3⁴ + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_2·b_3_5
       + c_4_19·b_1_2⁴ + c_4_21·a_1_1·b_3_11 + c_4_21·a_1_1·b_3_8 + c_4_20·a_1_1·b_3_11
       + c_4_20·a_1_1·b_3_8 + c_4_20·a_1_1·b_3_6 + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_11
       + c_4_19·a_1_1·b_3_8 + c_4_19·a_1_1·b_3_6 + c_4_19·a_1_1·b_1_3³ + c_4_19·a_1_0·b_3_5
       + c_4_21·a_1_1·a_3_1 + c_4_21·a_1_0·a_3_1
58. a_3_1·b_5_31 + a_1_1·b_1_3·b_3_6·b_3_8 + a_1_1·b_1_3²·b_5_32 + a_1_1·b_1_3⁴·b_3_11
       + a_1_1·b_1_3⁴·b_3_6 + c_4_21·a_1_1·b_1_3³ + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_8
       + c_4_20·a_1_1·b_1_3³ + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_6
       + c_4_19·a_1_1·b_1_3³ + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5
       + c_4_21·a_1_0·a_3_1 + c_4_20·a_1_0·a_3_1
59. b_3_11·b_5_31 + b_3_8·b_5_31 + b_3_6·b_5_31 + b_1_3³·b_5_32 + b_1_3⁵·b_3_6
       + b_1_2³·b_5_31 + b_1_2⁵·b_3_10 + b_1_2⁵·b_3_9 + b_1_2⁵·b_3_5 + b_1_2⁸
       + a_1_1·b_1_3⁴·b_3_8 + a_1_0³·b_5_31 + c_4_21·b_1_3·b_3_11 + c_4_21·b_1_3·b_3_8
       + c_4_21·b_1_2·b_3_10 + c_4_21·b_1_2·b_3_9 + c_4_21·b_1_2·b_3_5 + c_4_21·b_1_2⁴
       + c_4_20·b_1_3⁴ + c_4_20·b_1_2·b_3_5 + c_4_19·b_1_3·b_3_11 + c_4_19·b_1_3·b_3_8
       + c_4_19·b_1_3·b_3_6 + c_4_19·b_1_3⁴ + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_2·b_3_9
       + c_4_19·b_1_2·b_3_5 + c_4_19·b_1_2⁴ + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_9
       + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_1·b_1_3³ + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8
       + c_4_20·a_1_0·b_3_6 + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_1·b_3_11 + c_4_19·a_1_1·b_3_8
       + c_4_19·a_1_1·b_3_6 + c_4_19·a_1_1·b_1_3³ + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8
       + c_4_19·a_1_0·b_3_5 + c_4_20·a_1_1·a_3_1 + c_4_19·a_1_1·a_3_1 + c_4_19·a_1_0·a_3_1
60. b_5_32² + b_5_31·b_5_32 + b_1_3⁵·b_5_31 + b_1_3⁷·b_3_8 + b_1_2⁵·b_5_32
       + b_1_2⁵·b_5_31 + b_1_2⁷·b_3_10 + b_1_2⁷·b_3_9 + a_1_1·b_1_3⁶·b_3_11
       + a_1_1·b_1_3⁶·b_3_8 + c_4_21·b_1_3³·b_3_8 + c_4_21·b_1_3⁶ + c_4_21·b_1_2·b_5_32
       + c_4_21·b_1_2³·b_3_10 + c_4_21·b_1_2³·b_3_9 + c_4_21·b_1_2³·b_3_5
       + c_4_20·b_3_5·b_3_9 + c_4_20·b_3_5·b_3_8 + c_4_20·b_1_3·b_5_31 + c_4_20·b_1_2·b_5_31
       + c_4_20·b_1_2³·b_3_5 + c_4_20·b_1_2⁶ + c_4_19·b_3_6·b_3_8 + c_4_19·b_1_3·b_5_31
       + c_4_19·b_1_3³·b_3_11 + c_4_19·b_1_3³·b_3_8 + c_4_19·b_1_3³·b_3_6 + c_4_19·b_1_3⁶
       + c_4_19·b_1_2·b_5_32 + c_4_19·b_1_2³·b_3_10 + c_4_19·b_1_2³·b_3_9 + c_4_19·b_1_2⁶
       + c_4_21·a_1_1·b_5_32 + c_4_21·a_1_1·b_5_31 + c_4_21·a_1_1·b_1_3²·b_3_6
       + c_4_21·a_1_1·b_1_3⁵ + c_4_21·a_1_0·b_5_31 + c_4_20·a_1_0·b_5_32
       + c_4_20·a_1_0·b_5_31 + c_4_19·a_1_1·b_1_3²·b_3_8 + c_4_19·a_1_0·b_5_32
       + c_4_20·c_4_21·b_1_2² + c_4_20²·b_1_3² + c_4_21²·a_1_0·b_1_2
       + c_4_20·c_4_21·a_1_0·b_1_2 + c_4_19·c_4_21·a_1_1·b_1_3 + c_4_19·c_4_20·a_1_1·b_1_3
       + c_4_19²·a_1_0·b_1_2 + c_4_21²·a_1_1² + c_4_21²·a_1_0·a_1_1
       + c_4_20·c_4_21·a_1_0·a_1_1 + c_4_20·c_4_21·a_1_0² + c_4_20²·a_1_0·a_1_1
       + c_4_20²·a_1_0² + c_4_19·c_4_20·a_1_1² + c_4_19·c_4_20·a_1_0·a_1_1
       + c_4_19·c_4_20·a_1_0² + c_4_19²·a_1_0·a_1_1 + c_4_19²·a_1_0²
61. b_5_32² + b_5_31² + b_1_3⁴·b_3_6·b_3_8 + b_1_3⁷·b_3_8 + b_1_3⁷·b_3_6
       + b_1_2⁵·b_5_31 + b_1_2⁷·b_3_5 + b_1_2¹⁰ + a_1_1·b_1_3³·b_3_6·b_3_8
       + a_1_1·b_1_3⁴·b_5_32 + a_1_1·b_1_3⁶·b_3_11 + a_1_1·b_1_3⁶·b_3_6
       + c_4_20·b_1_2³·b_3_5 + c_4_20·b_1_2⁶ + c_4_19·b_1_3³·b_3_6 + c_4_19·b_1_3⁶
       + c_4_19·b_1_2³·b_3_10 + c_4_19·b_1_2³·b_3_9 + c_4_19·b_1_2³·b_3_5
       + c_4_21·a_1_1·b_1_3⁵ + c_4_20·a_1_1·b_1_3⁵ + c_4_19·a_1_1·b_1_3²·b_3_11
       + c_4_19·a_1_1·b_1_3⁵ + c_4_21²·b_1_2² + c_4_20²·b_1_3²
       + c_4_19·c_4_20·b_1_2² + c_4_19²·b_1_2² + c_4_21²·a_1_0² + c_4_20²·a_1_1²
       + c_4_20²·a_1_0² + c_4_19·c_4_20·a_1_0² + c_4_19²·a_1_0²
62. b_5_31² + b_1_3⁴·b_3_6·b_3_8 + b_1_3⁵·b_5_32 + b_1_2⁷·b_3_10 + b_1_2⁷·b_3_9
       + b_1_2⁷·b_3_5 + b_1_2¹⁰ + a_1_1·b_1_3³·b_3_6·b_3_8 + a_1_1·b_1_3⁶·b_3_6
       + c_4_21·b_1_3³·b_3_8 + c_4_21·b_1_3³·b_3_6 + c_4_21·b_1_2³·b_3_10
       + c_4_21·b_1_2³·b_3_9 + c_4_21·b_1_2³·b_3_5 + c_4_20·b_1_3⁶ + c_4_20·b_1_2³·b_3_10
       + c_4_20·b_1_2³·b_3_9 + c_4_20·b_1_2³·b_3_5 + c_4_20·b_1_2⁶ + c_4_19·b_1_3³·b_3_11
       + c_4_19·b_1_3⁶ + c_4_21·a_1_1·b_1_3²·b_3_6 + c_4_21·a_1_1·b_1_3⁵
       + c_4_20·a_1_1·b_1_3⁵ + c_4_21²·b_1_2² + c_4_20·c_4_21·b_1_2²
       + c_4_19·c_4_21·b_1_3² + c_4_19²·b_1_3² + c_4_19²·b_1_2² + c_4_21²·a_1_1²
       + c_4_20·c_4_21·a_1_0² + c_4_20²·a_1_0² + c_4_19·c_4_20·a_1_1²

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Ring generators

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• 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_19, a Duflot regular element of degree 4
  2. c_4_20, a Duflot regular element of degree 4
  3. c_4_21, a Duflot regular element of degree 4
  4. b_1_3² + b_1_2², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 10].
• 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

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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. a_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. a_3_1 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_6 → 0, an element of degree 3
8. b_3_8 → 0, an element of degree 3
9. b_3_9 → 0, an element of degree 3
10. b_3_10 → 0, an element of degree 3
11. b_3_11 → 0, an element of degree 3
12. c_4_19 → c_1_2⁴ + c_1_1⁴, an element of degree 4
13. c_4_20 → c_1_1⁴ + c_1_0⁴, an element of degree 4
14. c_4_21 → c_1_1⁴, an element of degree 4
15. b_5_31 → 0, an element of degree 5
16. b_5_32 → 0, an element of degree 5

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

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, 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. a_3_1 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_6 → c_1_2²·c_1_3, an element of degree 3
8. b_3_8 → c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
9. b_3_9 → c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
10. b_3_10 → c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
11. b_3_11 → c_1_2·c_1_3², an element of degree 3
12. c_4_19 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1·c_1_3³ + c_1_1⁴, an element of degree 4
13. c_4_20 → c_1_2²·c_1_3² + c_1_1·c_1_3³ + c_1_1⁴ + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
14. c_4_21 → c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1²·c_1_3² + c_1_1⁴, an element of degree 4
15. b_5_31 → c_1_2·c_1_3⁴ + c_1_2²·c_1_3³ + c_1_2³·c_1_3² + c_1_2⁴·c_1_3
       + c_1_1·c_1_2²·c_1_3² + c_1_1²·c_1_2²·c_1_3, an element of degree 5
16. b_5_32 → c_1_2²·c_1_3³ + c_1_2⁴·c_1_3 + c_1_1·c_1_2·c_1_3³ + c_1_1·c_1_2²·c_1_3²
       + c_1_1²·c_1_3³ + c_1_1²·c_1_2·c_1_3² + c_1_1²·c_1_2²·c_1_3 + c_1_1⁴·c_1_3
       + c_1_0²·c_1_3³ + c_1_0⁴·c_1_3, an element of degree 5

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

1. a_1_0 → 0, an element of degree 1
2. a_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. a_3_1 → 0, an element of degree 3
6. b_3_5 → c_1_3³ + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
7. b_3_6 → c_1_3³, an element of degree 3
8. b_3_8 → c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
9. b_3_9 → c_1_3³ + c_1_2·c_1_3² + c_1_2²·c_1_3 + 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
10. b_3_10 → c_1_3³ + c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
11. b_3_11 → c_1_0·c_1_3² + c_1_0²·c_1_3, an element of degree 3
12. c_4_19 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1·c_1_3³ + c_1_1⁴, an element of degree 4
13. c_4_20 → c_1_1·c_1_3³ + c_1_1⁴ + c_1_0·c_1_3³ + c_1_0⁴, an element of degree 4
14. c_4_21 → c_1_1·c_1_3³ + c_1_1⁴ + c_1_0·c_1_3³ + c_1_0²·c_1_3², an element of degree 4
15. b_5_31 → c_1_2²·c_1_3³ + c_1_2⁴·c_1_3 + c_1_1²·c_1_3³ + c_1_1⁴·c_1_3
       + c_1_0·c_1_1·c_1_3³ + c_1_0·c_1_1²·c_1_3² + c_1_0²·c_1_1·c_1_3²
       + c_1_0²·c_1_1²·c_1_3, an element of degree 5
16. b_5_32 → c_1_1·c_1_3⁴ + c_1_1·c_1_2·c_1_3³ + c_1_1·c_1_2²·c_1_3² + c_1_1²·c_1_3³
       + c_1_1²·c_1_2·c_1_3² + c_1_1²·c_1_2²·c_1_3 + c_1_0·c_1_3⁴
       + c_1_0·c_1_2·c_1_3³ + c_1_0·c_1_2²·c_1_3² + c_1_0²·c_1_3³
       + c_1_0²·c_1_2·c_1_3² + c_1_0²·c_1_2²·c_1_3, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128