Cohomology of group number 1934 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 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) · (t9  +  2·t8  +  2·t6  +  2·t5  +  2·t4  +  2·t3  +  2·t2  +  2·t  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1)2 · (t4  +  1)
  • The a-invariants are -∞,-∞,-5,-3. 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 14 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_2, a nilpotent 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. a_3_11, a nilpotent element of degree 3
  7. b_4_15, an element of degree 4
  8. c_4_16, a Duflot regular element of degree 4
  9. b_5_22, an element of degree 5
  10. b_5_23, an element of degree 5
  11. b_6_31, an element of degree 6
  12. a_8_32, a nilpotent element of degree 8
  13. b_8_43, an element of degree 8
  14. c_8_45, a Duflot regular element of degree 8

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

Ring relations

There are 57 minimal relations of maximal degree 16:

  1. a_1_0·a_1_2
  2. a_1_2·b_1_1 + a_1_0·b_1_3 + a_1_0·b_1_1
  3. b_1_1·b_1_32 + b_1_12·b_1_3 + a_1_23 + a_1_03
  4. a_1_0·b_1_32 + a_1_0·b_1_1·b_1_3
  5. a_1_2·a_3_10
  6. b_1_3·a_3_10 + b_1_1·a_3_11 + b_1_1·a_3_10
  7. a_1_0·a_3_11
  8. a_1_23·b_1_32 + a_1_03·b_1_12
  9. b_4_15·b_1_3 + a_1_22·b_1_33 + a_1_22·a_3_11 + a_1_02·a_3_10
  10. b_1_12·a_3_11 + b_4_15·a_1_0
  11. b_1_1·b_1_3·a_3_11 + b_1_12·a_3_11 + b_4_15·a_1_2 + a_1_03·b_1_12
  12. a_3_10·a_3_11
  13. a_3_102 + a_1_0·b_1_12·a_3_10 + a_1_02·b_1_14 + a_1_02·b_1_1·a_3_10
       + a_1_03·a_3_10 + c_4_16·a_1_02
  14. a_3_112 + a_1_2·b_1_32·a_3_11 + a_1_22·b_1_34 + a_1_22·b_1_3·a_3_11
       + c_4_16·a_1_22
  15. a_1_2·b_5_22 + b_4_15·a_1_0·b_1_1 + a_3_112 + a_1_2·b_1_32·a_3_11 + a_1_22·b_1_34
       + a_1_23·a_3_11
  16. b_1_3·b_5_22 + b_1_1·b_5_23 + b_1_16 + a_1_0·b_1_14·b_1_3 + a_1_2·b_1_32·a_3_11
       + a_1_0·b_1_12·a_3_10 + a_1_03·a_3_10 + c_4_16·a_1_2·b_1_3
  17. a_1_0·b_5_23 + a_1_0·b_5_22 + a_1_0·b_1_15 + b_4_15·a_1_0·b_1_1 + a_1_02·b_1_14
       + a_1_02·b_1_1·a_3_10 + a_1_23·a_3_11
  18. b_4_15·a_3_11 + b_4_15·a_1_0·b_1_12 + a_1_22·b_1_32·a_3_11 + a_1_03·b_1_1·a_3_10
       + c_4_16·a_1_0·b_1_1·b_1_3 + c_4_16·a_1_0·b_1_12
  19. b_4_15·a_3_10 + b_4_15·a_1_0·b_1_12 + a_1_03·b_1_1·a_3_10
       + c_4_16·a_1_0·b_1_1·b_1_3 + c_4_16·a_1_0·b_1_12
  20. b_1_1·b_1_3·b_5_23 + b_1_12·b_5_23 + b_1_16·b_1_3 + b_1_17 + a_1_23·b_1_3·a_3_11
       + a_1_03·b_1_1·a_3_10
  21. b_1_1·b_1_3·b_5_23 + b_1_16·b_1_3 + b_6_31·b_1_3 + a_1_0·b_1_1·b_5_22
       + b_4_15·a_1_0·b_1_12 + a_1_22·b_5_23 + a_1_22·b_1_35 + a_1_0·b_1_13·a_3_10
       + a_1_02·b_5_22 + a_1_02·b_1_15 + c_4_16·b_1_12·b_1_3 + c_4_16·a_1_2·b_1_32
       + c_4_16·a_1_02·b_1_1 + c_4_16·a_1_23 + c_4_16·a_1_03
  22. b_1_1·b_1_3·b_5_23 + b_1_12·b_5_22 + b_1_16·b_1_3 + b_4_15·b_1_13
       + a_1_0·b_1_1·b_5_22 + a_1_0·b_1_15·b_1_3 + b_6_31·a_1_0 + b_4_15·a_1_0·b_1_12
       + a_1_0·b_1_13·a_3_10 + a_1_02·b_5_22 + a_1_02·b_1_15 + a_1_02·b_1_12·a_3_10
       + a_1_23·b_1_3·a_3_11 + c_4_16·a_1_0·b_1_12 + c_4_16·a_1_03
  23. b_1_1·b_1_3·b_5_23 + b_1_12·b_5_22 + b_1_16·b_1_3 + b_4_15·b_1_13
       + a_1_0·b_1_15·b_1_3 + b_6_31·a_1_2 + a_1_0·b_1_13·a_3_10 + a_1_02·b_1_12·a_3_10
       + a_1_23·b_1_3·a_3_11 + c_4_16·a_1_0·b_1_1·b_1_3 + c_4_16·a_1_0·b_1_12
       + c_4_16·a_1_22·b_1_3
  24. b_4_15·b_1_14 + b_4_152 + b_4_15·a_1_0·b_1_13 + a_1_02·b_1_13·a_3_10
       + c_4_16·b_1_13·b_1_3 + c_4_16·b_1_14 + c_4_16·a_1_03·b_1_1
  25. a_3_11·b_5_22 + b_4_15·a_1_0·b_1_13 + a_1_22·b_1_33·a_3_11
       + c_4_16·a_1_0·b_1_12·b_1_3 + c_4_16·a_1_0·b_1_13 + c_4_16·a_1_2·a_3_11
       + c_4_16·a_1_23·b_1_3
  26. a_3_10·b_5_23 + a_3_10·b_5_22 + b_1_15·a_3_10 + b_4_15·a_1_0·b_1_13
       + a_1_0·b_1_14·a_3_10 + a_1_02·b_1_13·a_3_10 + c_4_16·a_1_0·b_1_12·b_1_3
       + c_4_16·a_1_0·b_1_13 + c_4_16·a_1_03·b_1_1
  27. b_4_15·b_5_23 + b_4_15·b_5_22 + b_6_31·a_3_11 + b_4_152·a_1_0 + a_1_22·b_1_32·b_5_23
       + a_1_22·b_1_34·a_3_11 + a_1_02·b_1_14·a_3_10 + c_4_16·b_1_14·b_1_3
       + c_4_16·b_1_15 + c_4_16·a_1_0·b_1_13·b_1_3 + c_4_16·a_1_0·b_1_14
       + b_4_15·c_4_16·a_1_0 + c_4_16·a_1_2·b_1_3·a_3_11
  28. b_4_15·b_5_22 + b_4_152·b_1_1 + b_1_1·a_3_10·b_5_22 + b_6_31·a_3_10 + b_4_152·a_1_0
       + a_1_0·a_3_10·b_5_22 + a_1_0·b_1_15·a_3_10 + a_1_02·b_1_14·a_3_10
       + c_4_16·b_1_12·a_3_10 + b_4_15·c_4_16·a_1_2 + b_4_15·c_4_16·a_1_0
       + c_4_16·a_1_02·a_3_10
  29. b_4_15·b_5_23 + b_4_152·b_1_1 + b_1_36·a_3_11 + b_1_1·a_3_10·b_5_22 + b_1_16·a_3_10
       + a_1_2·b_1_33·b_5_23 + a_8_32·b_1_3 + b_6_31·a_1_0·b_1_1·b_1_3 + b_4_152·a_1_0
       + a_1_2·b_1_35·a_3_11 + a_1_22·b_1_32·b_5_23 + a_1_0·b_1_15·a_3_10
       + b_6_31·a_1_02·b_1_1 + a_1_22·b_1_34·a_3_11 + c_4_16·b_1_14·b_1_3
       + c_4_16·b_1_15 + c_4_16·b_1_12·a_3_10 + b_4_15·c_4_16·a_1_2
       + c_4_16·a_1_2·b_1_3·a_3_11 + c_4_16·a_1_22·b_1_33 + c_4_16·a_1_0·b_1_1·a_3_10
       + c_4_16·a_1_02·b_1_13 + c_4_16·a_1_03·b_1_12
  30. b_4_15·b_5_22 + b_4_152·b_1_1 + b_1_1·a_3_10·b_5_22 + b_1_16·a_3_10 + a_8_32·b_1_1
       + b_6_31·a_1_0·b_1_1·b_1_3 + b_4_152·a_1_0 + a_1_0·b_1_15·a_3_10
       + b_6_31·a_1_02·b_1_1 + a_1_02·b_1_14·a_3_10 + c_4_16·b_1_12·a_3_10
       + c_4_16·a_1_0·b_1_13·b_1_3 + c_4_16·a_1_0·b_1_14 + b_4_15·c_4_16·a_1_2
       + b_4_15·c_4_16·a_1_0 + c_4_16·a_1_0·b_1_1·a_3_10 + c_4_16·a_1_02·b_1_13
  31. a_1_0·a_3_10·b_5_22 + a_1_0·b_1_15·a_3_10 + a_8_32·a_1_0 + b_6_31·a_1_02·b_1_1
       + a_1_02·b_1_14·a_3_10 + c_4_16·a_1_0·b_1_1·a_3_10 + c_4_16·a_1_02·a_3_10
  32. a_1_2·b_1_35·a_3_11 + a_1_22·b_1_32·b_5_23 + a_8_32·a_1_2 + a_1_22·b_1_34·a_3_11
       + c_4_16·a_1_22·a_3_11 + c_4_16·a_1_03·b_1_12
  33. b_1_34·b_5_23 + b_1_18·b_1_3 + b_8_43·b_1_3 + b_6_31·b_1_12·b_1_3
       + a_1_2·b_1_33·b_5_23 + a_1_0·b_1_17·b_1_3 + b_6_31·a_1_0·b_1_1·b_1_3
       + a_1_2·a_3_11·b_5_23 + a_1_0·a_3_10·b_5_22 + a_1_02·b_1_17 + a_1_02·b_1_14·a_3_10
       + c_4_16·b_1_35 + b_4_15·c_4_16·a_1_2 + b_4_15·c_4_16·a_1_0 + c_4_16·a_1_22·a_3_11
       + c_4_16·a_1_02·a_3_10 + c_4_16·a_1_03·b_1_12
  34. b_4_15·b_5_22 + b_4_152·b_1_1 + b_8_43·a_1_0 + a_1_02·b_1_17
       + a_1_02·b_1_14·a_3_10 + b_4_15·c_4_16·a_1_2 + b_4_15·c_4_16·a_1_0
       + c_4_16·a_1_02·b_1_13 + c_4_16·a_1_03·b_1_12
  35. b_4_15·b_5_23 + b_4_15·b_5_22 + a_1_2·b_1_33·b_5_23 + b_8_43·a_1_2 + b_4_152·a_1_0
       + a_1_02·b_1_14·a_3_10 + c_4_16·b_1_14·b_1_3 + c_4_16·b_1_15
       + c_4_16·a_1_2·b_1_34 + c_4_16·a_1_0·b_1_13·b_1_3 + c_4_16·a_1_0·b_1_14
  36. b_5_22·b_5_23 + b_1_19·b_1_3 + b_6_31·b_1_13·b_1_3 + b_4_152·b_1_12
       + a_1_0·b_1_18·b_1_3 + b_6_31·a_1_0·b_1_13 + b_4_152·a_1_0·b_1_1
       + a_1_2·b_1_3·a_3_11·b_5_23 + a_1_02·b_1_18 + a_8_32·a_1_0·b_1_1
       + b_6_31·a_1_02·b_1_12 + c_4_16·b_1_15·b_1_3 + c_4_16·b_1_16
       + c_4_16·a_1_2·b_5_23 + c_4_16·a_1_0·b_1_15 + c_8_45·a_1_02
       + c_4_16·a_1_0·b_1_12·a_3_10 + c_4_16·a_1_03·a_3_10
  37. b_5_22·b_5_23 + b_5_222 + b_6_31·b_1_13·b_1_3 + a_1_0·b_1_18·b_1_3
       + b_6_31·a_1_0·b_1_12·b_1_3 + b_6_31·a_1_0·b_1_13 + b_4_152·a_1_0·b_1_1
       + a_1_2·b_1_3·a_3_11·b_5_23 + a_1_02·b_1_18 + b_6_31·a_1_0·a_3_10
       + b_6_31·a_1_02·b_1_12 + a_8_32·a_1_02 + c_4_16·b_1_16 + c_4_16·a_1_2·b_5_23
       + c_4_16·a_1_0·b_1_14·b_1_3 + c_4_16·a_1_0·b_1_15 + c_4_16·a_1_0·b_1_12·a_3_10
       + c_4_16·a_1_02·b_1_1·a_3_10 + c_4_162·a_1_22
  38. b_5_222 + b_1_19·b_1_3 + b_8_43·b_1_12 + b_4_15·b_6_31 + b_4_152·b_1_12
       + a_1_0·b_1_19 + b_6_31·a_1_0·b_1_13 + a_1_02·b_1_18 + b_6_31·a_1_0·a_3_10
       + b_6_31·a_1_02·b_1_12 + a_1_02·b_1_15·a_3_10 + c_4_16·b_1_15·b_1_3
       + b_4_15·c_4_16·b_1_12 + c_4_16·a_1_0·b_1_14·b_1_3 + c_4_16·a_1_0·b_1_15
       + c_8_45·a_1_02 + c_4_16·a_1_0·b_1_12·a_3_10 + c_4_16·a_1_02·b_1_14
       + c_4_16·a_1_23·a_3_11 + c_4_16·a_1_03·a_3_10 + c_4_162·a_1_22
  39. b_5_232 + b_5_222 + b_1_110 + b_4_152·b_1_12 + b_8_43·a_1_2·b_1_3
       + a_1_02·b_1_18 + b_8_43·a_1_22 + a_8_32·a_1_22 + c_4_16·b_1_36
       + c_4_16·b_1_15·b_1_3 + c_4_16·a_1_2·b_1_35 + c_8_45·a_1_22
       + c_4_16·a_1_22·b_1_3·a_3_11 + c_4_162·a_1_22
  40. b_1_110·b_1_3 + b_6_31·b_5_22 + b_4_15·b_6_31·b_1_1 + a_1_0·b_1_19·b_1_3
       + b_4_15·b_6_31·a_1_0 + b_4_152·a_1_0·b_1_12 + a_8_32·a_1_0·b_1_12
       + b_6_31·a_1_0·b_1_1·a_3_10 + b_6_31·a_1_02·b_1_13 + c_4_16·b_1_16·b_1_3
       + c_4_16·b_6_31·b_1_3 + c_8_45·a_1_0·b_1_1·b_1_3 + c_8_45·a_1_0·b_1_12
       + c_4_16·a_1_0·b_1_15·b_1_3 + c_4_16·a_1_0·b_1_16 + c_4_16·b_6_31·a_1_0
       + b_4_15·c_4_16·a_1_0·b_1_12 + c_8_45·a_1_02·b_1_1 + c_4_16·a_1_22·b_5_23
       + c_4_16·a_1_22·b_1_35 + c_4_16·a_1_02·b_5_22 + c_8_45·a_1_03
       + c_4_16·a_1_22·b_1_32·a_3_11 + c_4_16·a_1_02·b_1_12·a_3_10
       + c_4_16·a_1_03·b_1_1·a_3_10 + c_4_162·b_1_12·b_1_3 + c_4_162·a_1_2·b_1_32
       + c_4_162·a_1_0·b_1_1·b_1_3 + c_4_162·a_1_22·b_1_3 + c_4_162·a_1_02·b_1_1
       + c_4_162·a_1_23
  41. b_6_31·b_5_23 + b_6_31·b_5_22 + b_6_31·b_1_15 + b_4_15·b_6_31·b_1_1
       + b_6_31·a_1_0·b_1_13·b_1_3 + b_4_15·b_6_31·a_1_0 + b_4_152·a_1_0·b_1_12
       + a_1_2·b_1_32·a_3_11·b_5_23 + a_1_22·b_1_39 + a_8_32·a_3_11
       + a_8_32·a_1_2·b_1_32 + b_6_31·a_1_0·b_1_1·a_3_10 + a_8_32·a_1_22·b_1_3
       + b_6_31·a_1_02·a_3_10 + c_8_45·a_1_0·b_1_1·b_1_3 + c_8_45·a_1_0·b_1_12
       + c_4_16·a_1_2·b_1_3·b_5_23 + c_4_16·a_1_0·b_1_15·b_1_3 + c_4_16·a_1_0·b_1_16
       + c_4_16·b_6_31·a_1_2 + b_4_15·c_4_16·a_1_0·b_1_12 + c_4_16·a_1_23·b_1_3·a_3_11
       + c_4_16·a_1_03·b_1_1·a_3_10 + c_4_162·a_1_23
  42. b_1_110·b_1_3 + b_6_31·b_5_22 + b_4_15·b_6_31·b_1_1 + a_1_0·b_1_19·b_1_3
       + b_4_15·b_6_31·a_1_0 + b_4_152·a_1_0·b_1_12 + a_1_02·b_1_19 + a_8_32·a_3_10
       + b_6_31·a_1_02·b_1_13 + a_8_32·a_1_02·b_1_1 + b_6_31·a_1_02·a_3_10
       + c_4_16·b_1_16·b_1_3 + c_4_16·b_6_31·b_1_3 + c_8_45·a_1_0·b_1_1·b_1_3
       + c_8_45·a_1_0·b_1_12 + c_4_16·a_1_0·b_1_15·b_1_3 + c_4_16·a_1_0·b_1_16
       + c_4_16·b_6_31·a_1_0 + b_4_15·c_4_16·a_1_0·b_1_12 + c_8_45·a_1_02·b_1_1
       + c_4_16·a_1_22·b_5_23 + c_4_16·a_1_22·b_1_35 + c_4_16·a_1_02·b_1_15
       + c_8_45·a_1_03 + c_4_16·a_1_22·b_1_32·a_3_11 + c_4_16·a_1_03·b_1_1·a_3_10
       + c_4_162·b_1_12·b_1_3 + c_4_162·a_1_2·b_1_32 + c_4_162·a_1_0·b_1_1·b_1_3
       + c_4_162·a_1_22·b_1_3 + c_4_162·a_1_23 + c_4_162·a_1_03
  43. b_6_31·b_5_23 + b_6_31·b_5_22 + b_6_31·b_1_15 + b_4_15·b_6_31·b_1_1
       + b_6_31·a_1_0·b_1_13·b_1_3 + b_4_15·b_6_31·a_1_0 + b_4_152·a_1_0·b_1_12
       + b_8_43·a_1_22·b_1_3 + b_6_31·a_1_0·b_1_1·a_3_10 + b_6_31·a_1_02·a_3_10
       + c_8_45·a_1_0·b_1_1·b_1_3 + c_8_45·a_1_0·b_1_12 + c_4_16·a_1_2·b_1_3·b_5_23
       + c_4_16·a_1_0·b_1_15·b_1_3 + c_4_16·a_1_0·b_1_16 + c_4_16·b_6_31·a_1_2
       + b_4_15·c_4_16·a_1_0·b_1_12 + c_4_16·a_1_22·b_1_32·a_3_11
       + c_4_16·a_1_23·b_1_3·a_3_11 + c_4_16·a_1_03·b_1_1·a_3_10
  44. b_1_33·a_3_11·b_5_23 + b_8_43·a_3_11 + b_4_15·b_6_31·a_1_0
       + a_1_2·b_1_32·a_3_11·b_5_23 + a_1_22·b_1_3·a_3_11·b_5_23 + c_4_16·b_1_34·a_3_11
       + c_4_16·b_6_31·a_1_2 + b_4_15·c_4_16·a_1_0·b_1_12 + c_4_16·a_1_23·b_1_3·a_3_11
       + c_4_162·a_1_0·b_1_1·b_1_3 + c_4_162·a_1_0·b_1_12 + c_4_162·a_1_22·b_1_3
  45. b_1_110·b_1_3 + b_6_31·b_5_22 + b_4_15·b_6_31·b_1_1 + a_1_0·b_1_19·b_1_3
       + b_8_43·a_3_10 + b_4_152·a_1_0·b_1_12 + c_4_16·b_1_16·b_1_3 + c_4_16·b_6_31·b_1_3
       + c_8_45·a_1_0·b_1_1·b_1_3 + c_8_45·a_1_0·b_1_12 + c_4_16·a_1_0·b_1_15·b_1_3
       + c_4_16·a_1_0·b_1_16 + c_4_16·b_6_31·a_1_2 + c_4_16·b_6_31·a_1_0
       + c_8_45·a_1_02·b_1_1 + c_4_16·a_1_22·b_5_23 + c_4_16·a_1_22·b_1_35
       + c_4_16·a_1_0·b_1_13·a_3_10 + c_4_16·a_1_02·b_5_22 + c_8_45·a_1_03
       + c_4_16·a_1_22·b_1_32·a_3_11 + c_4_16·a_1_02·b_1_12·a_3_10
       + c_4_16·a_1_23·b_1_3·a_3_11 + c_4_162·b_1_12·b_1_3 + c_4_162·a_1_2·b_1_32
       + c_4_162·a_1_0·b_1_12 + c_4_162·a_1_02·b_1_1 + c_4_162·a_1_23
  46. b_6_31·b_1_1·b_5_22 + b_6_312 + b_4_153 + a_1_0·b_1_111
       + b_6_31·a_1_0·b_1_14·b_1_3 + b_6_31·a_1_02·b_1_14 + a_8_32·a_1_02·b_1_12
       + b_6_31·a_1_02·b_1_1·a_3_10 + c_8_45·b_1_13·b_1_3 + c_8_45·b_1_14
       + c_4_16·b_1_17·b_1_3 + c_4_16·b_1_18 + c_4_16·b_6_31·b_1_1·b_1_3
       + c_8_45·a_1_0·b_1_12·b_1_3 + c_8_45·a_1_0·b_1_13 + c_4_16·a_1_0·b_1_17
       + c_4_16·b_6_31·a_1_0·b_1_1 + b_4_15·c_4_16·a_1_0·b_1_13 + c_4_16·a_1_02·b_1_16
       + c_4_16·b_6_31·a_1_02 + c_4_162·a_1_0·b_1_12·b_1_3 + c_4_162·a_1_22·b_1_32
       + c_4_162·a_1_03·b_1_1
  47. b_4_15·a_8_32 + b_4_15·b_6_31·a_1_0·b_1_1 + b_4_152·a_1_0·b_1_13
       + a_8_32·a_1_22·b_1_32 + c_4_16·b_6_31·a_1_0·b_1_3 + c_4_16·b_6_31·a_1_0·b_1_1
       + b_4_15·c_4_16·a_1_0·b_1_13
  48. b_4_15·b_8_43 + b_4_15·b_6_31·b_1_12 + b_4_152·a_1_0·b_1_13
       + b_8_43·a_1_22·b_1_32 + a_8_32·a_1_2·a_3_11 + a_8_32·a_1_22·b_1_32
       + b_6_31·a_1_02·b_1_1·a_3_10 + c_4_16·b_6_31·b_1_1·b_1_3 + c_4_16·b_6_31·b_1_12
       + b_4_152·c_4_16 + c_4_16·a_1_0·b_1_16·b_1_3 + c_4_16·a_1_0·b_1_17
       + b_4_15·c_4_16·a_1_0·b_1_13 + c_4_16·a_1_22·b_1_33·a_3_11
       + c_4_162·a_1_03·b_1_1
  49. a_8_32·b_5_22 + a_8_32·b_1_15 + b_6_31·a_1_0·b_1_15·b_1_3 + b_6_312·a_1_0
       + b_4_15·b_6_31·a_1_0·b_1_12 + b_4_153·a_1_0 + a_8_32·a_1_0·b_1_14
       + b_6_31·a_1_0·b_1_13·a_3_10 + b_6_31·a_1_02·b_1_15 + a_8_32·a_1_2·b_1_3·a_3_11
       + b_6_31·a_1_02·b_1_12·a_3_10 + c_8_45·a_1_0·b_1_13·b_1_3 + c_8_45·a_1_0·b_1_14
       + c_4_16·a_1_0·b_1_17·b_1_3 + c_4_16·a_1_0·b_1_18 + c_4_16·b_8_43·a_1_0
       + c_4_16·b_6_31·a_3_10 + c_4_16·b_6_31·a_1_0·b_1_12 + b_4_152·c_4_16·a_1_0
       + c_4_16·a_8_32·a_1_2 + c_4_16·b_6_31·a_1_02·b_1_1 + c_8_45·a_1_02·a_3_10
       + c_4_162·b_1_12·a_3_10 + c_4_162·a_1_0·b_1_13·b_1_3 + c_4_162·a_1_0·b_1_14
       + c_4_162·a_1_02·a_3_10
  50. b_8_43·b_5_23 + b_4_15·b_6_31·b_1_13 + a_1_0·b_1_112 + b_8_43·a_1_2·b_1_34
       + a_8_32·b_5_22 + a_8_32·b_1_15 + b_6_31·a_1_0·b_1_15·b_1_3 + b_6_31·a_1_0·b_1_16
       + b_6_312·a_1_0 + b_4_15·b_6_31·a_1_0·b_1_12 + b_4_153·a_1_0
       + a_8_32·a_1_0·b_1_14 + b_6_31·a_1_0·b_1_13·a_3_10 + b_6_31·a_1_02·b_5_22
       + b_6_31·a_1_02·b_1_15 + a_8_32·a_1_02·b_1_13 + c_4_16·b_1_39
       + c_4_16·b_1_18·b_1_3 + c_4_16·b_8_43·b_1_3 + b_4_152·c_4_16·b_1_1
       + c_8_45·a_1_0·b_1_13·b_1_3 + c_8_45·a_1_0·b_1_14 + c_4_16·a_1_0·b_1_17·b_1_3
       + c_4_16·b_8_43·a_1_2 + c_4_16·b_6_31·a_3_10 + c_4_16·b_6_31·a_1_0·b_1_1·b_1_3
       + c_4_16·b_6_31·a_1_0·b_1_12 + b_4_15·c_8_45·a_1_2 + b_4_15·c_8_45·a_1_0
       + c_8_45·a_1_22·b_1_33 + c_4_16·a_1_2·a_3_11·b_5_23 + c_4_16·a_1_22·b_1_37
       + c_4_16·a_1_0·b_1_15·a_3_10 + c_4_16·a_8_32·a_1_0 + c_4_16·b_6_31·a_1_02·b_1_1
       + c_8_45·a_1_02·a_3_10 + c_4_162·b_1_35 + c_4_162·b_1_15
       + c_4_162·b_1_12·a_3_10 + c_4_162·a_1_2·b_1_34 + c_4_162·a_1_0·b_1_1·a_3_10
       + c_4_162·a_1_02·b_1_13 + c_4_162·a_1_02·a_3_10
  51. b_8_43·b_1_32·a_3_11 + a_8_32·b_5_23 + a_8_32·b_5_22 + a_8_32·b_1_15
       + b_4_15·b_6_31·a_1_0·b_1_12 + b_4_153·a_1_0 + b_8_43·a_1_22·b_1_33
       + a_8_32·a_1_0·b_1_14 + a_8_32·a_1_2·b_1_3·a_3_11 + a_8_32·a_1_02·b_1_13
       + b_6_31·a_1_02·b_1_12·a_3_10 + c_4_16·a_1_2·b_1_38 + c_4_16·b_8_43·a_1_2
       + c_4_16·a_8_32·b_1_3 + c_4_16·a_8_32·b_1_1 + c_4_16·b_6_31·a_1_0·b_1_1·b_1_3
       + c_4_16·b_6_31·a_1_0·b_1_12 + b_4_152·c_4_16·a_1_0 + c_4_16·a_1_2·a_3_11·b_5_23
       + c_4_16·a_1_22·b_1_32·b_5_23 + c_4_16·a_1_22·b_1_37 + c_8_45·a_1_03·b_1_12
       + c_4_16·a_1_22·b_1_34·a_3_11 + c_4_162·a_1_2·b_1_34 + b_4_15·c_4_162·a_1_0
       + c_4_162·a_1_2·b_1_3·a_3_11 + c_4_162·a_1_22·b_1_33 + c_4_162·a_1_22·a_3_11
       + c_4_162·a_1_03·b_1_12
  52. b_8_43·b_5_22 + b_4_15·b_6_31·b_1_13 + a_8_32·b_5_22 + a_8_32·b_1_15
       + b_6_312·a_1_0 + b_4_153·a_1_0 + b_8_43·a_1_2·b_1_3·a_3_11
       + b_6_31·a_1_0·b_1_13·a_3_10 + b_6_31·a_1_02·b_5_22 + a_8_32·a_1_2·b_1_3·a_3_11
       + a_8_32·a_1_02·b_1_13 + b_6_31·a_1_02·b_1_12·a_3_10
       + c_4_16·b_6_31·b_1_12·b_1_3 + c_4_16·b_6_31·b_1_13 + b_4_152·c_4_16·b_1_1
       + c_8_45·a_1_0·b_1_13·b_1_3 + c_8_45·a_1_0·b_1_14 + c_4_16·a_1_0·b_1_17·b_1_3
       + c_4_16·b_8_43·a_1_2 + c_4_16·b_6_31·a_3_10 + b_4_15·c_8_45·a_1_0
       + b_4_152·c_4_16·a_1_0 + c_4_16·a_1_02·b_1_17 + c_4_16·a_8_32·a_1_2
       + c_8_45·a_1_02·a_3_10 + c_4_162·b_1_12·a_3_10 + c_4_162·a_1_0·b_1_13·b_1_3
       + b_4_15·c_4_162·a_1_0 + c_4_162·a_1_02·a_3_10 + c_4_162·a_1_03·b_1_12
  53. b_6_31·a_3_10·b_5_22 + b_6_31·b_1_15·a_3_10 + b_6_31·a_8_32 + b_6_312·a_1_0·b_1_1
       + b_4_15·b_6_31·a_1_0·b_1_13 + b_6_31·a_1_0·b_1_14·a_3_10 + b_6_312·a_1_02
       + a_8_32·a_1_2·b_1_32·a_3_11 + a_8_32·a_1_02·b_1_14 + c_8_45·a_1_0·b_1_14·b_1_3
       + c_8_45·a_1_0·b_1_15 + c_4_16·a_1_0·b_1_18·b_1_3 + c_4_16·a_1_0·b_1_19
       + c_4_16·b_6_31·b_1_1·a_3_11 + c_4_16·b_6_31·b_1_1·a_3_10
       + c_4_16·b_6_31·a_1_0·b_1_12·b_1_3 + c_4_16·b_6_31·a_1_0·b_1_13
       + b_4_15·c_8_45·a_1_0·b_1_1 + c_4_16·a_8_32·a_1_2·b_1_3 + c_4_16·b_6_31·a_1_0·a_3_10
       + c_4_16·b_6_31·a_1_02·b_1_12 + c_4_16·a_8_32·a_1_02 + c_8_45·a_1_03·a_3_10
       + c_4_162·a_1_0·b_1_14·b_1_3 + c_4_162·a_1_0·b_1_15
       + c_4_162·a_1_02·b_1_1·a_3_10 + c_4_162·a_1_03·a_3_10
  54. b_6_31·b_8_43 + b_4_152·b_6_31 + b_4_153·b_1_12 + b_6_31·a_1_0·b_1_17
       + b_8_43·a_1_22·b_1_34 + a_8_32·a_1_0·b_1_15 + a_8_32·a_1_02·b_1_14
       + b_6_31·a_1_02·b_1_13·a_3_10 + b_4_15·c_8_45·b_1_12 + b_4_15·c_4_16·b_6_31
       + c_8_45·a_1_0·b_1_14·b_1_3 + c_8_45·a_1_0·b_1_15 + c_4_16·b_8_43·a_1_2·b_1_3
       + c_4_16·b_6_31·a_1_0·b_1_12·b_1_3 + c_4_16·a_1_22·b_1_38
       + c_4_16·b_8_43·a_1_22 + c_4_16·a_8_32·a_1_0·b_1_1 + c_8_45·a_1_02·b_1_1·a_3_10
       + c_8_45·a_1_03·a_3_10 + c_4_162·b_1_15·b_1_3 + c_4_162·b_1_16
       + c_4_162·a_1_22·b_1_34 + c_4_162·a_1_0·b_1_12·a_3_10
       + c_4_162·a_1_02·b_1_1·a_3_10 + c_4_162·a_1_23·a_3_11 + c_4_162·a_1_03·a_3_10
  55. a_8_32·b_1_35·a_3_11 + a_8_32·a_1_2·b_1_32·b_5_23 + a_8_322
       + b_6_312·a_1_02·b_1_12 + a_8_32·a_1_22·b_1_3·b_5_23 + a_8_32·a_1_22·b_1_36
       + b_6_31·a_1_02·b_1_15·a_3_10 + c_4_16·a_8_32·b_1_1·a_3_10
       + c_4_16·b_6_31·a_1_02·b_1_14 + c_4_16·a_8_32·a_1_2·a_3_11
       + c_4_16·a_8_32·a_1_22·b_1_32 + c_4_16·a_8_32·a_1_02·b_1_12
       + c_4_162·a_1_0·b_1_14·a_3_10 + c_4_162·a_1_02·b_1_16
       + c_4_162·b_6_31·a_1_02 + c_4_163·a_1_02·b_1_12 + c_4_163·a_1_03·b_1_1
  56. b_8_432 + b_4_152·b_6_31·b_1_12 + b_4_154 + b_8_43·a_1_2·b_1_37
       + b_4_153·a_1_0·b_1_13 + a_8_32·a_3_11·b_5_23 + a_8_32·a_1_2·b_1_32·b_5_23
       + b_6_312·a_1_02·b_1_12 + a_8_32·a_1_22·b_1_36 + c_4_16·b_1_312
       + c_4_16·b_6_312 + b_4_152·c_8_45 + b_4_153·c_4_16 + c_4_16·a_1_2·b_1_311
       + c_4_16·a_1_0·b_1_111 + c_4_16·b_6_31·a_1_0·b_5_22
       + c_4_16·b_6_31·a_1_0·b_1_14·b_1_3 + b_4_15·c_4_16·b_6_31·a_1_0·b_1_1
       + b_4_152·c_4_16·a_1_0·b_1_13 + c_8_45·a_1_22·b_1_36
       + c_4_16·b_8_43·a_1_2·a_3_11 + c_4_16·b_8_43·a_1_22·b_1_32
       + c_4_16·a_8_32·b_1_3·a_3_11 + c_4_16·a_8_32·a_1_0·b_1_13
       + c_4_16·b_6_31·a_1_0·b_1_12·a_3_10 + c_4_16·a_8_32·a_1_22·b_1_32
       + c_4_16·b_6_31·a_1_02·b_1_1·a_3_10 + c_4_16·c_8_45·b_1_13·b_1_3
       + c_4_16·c_8_45·b_1_14 + c_4_162·b_1_38 + c_4_162·b_1_17·b_1_3
       + c_4_162·b_1_18 + c_4_162·a_1_0·b_1_16·b_1_3 + c_4_162·a_1_0·b_1_17
       + c_4_162·b_6_31·a_1_0·b_1_3 + b_4_15·c_4_162·a_1_0·b_1_13
       + c_4_16·c_8_45·a_1_02·b_1_12 + c_4_162·a_1_2·b_1_34·a_3_11
       + c_4_162·a_1_02·b_1_16 + c_4_162·b_6_31·a_1_02
       + c_4_162·a_1_02·b_1_13·a_3_10 + c_4_163·b_1_14 + c_4_163·a_1_0·b_1_12·b_1_3
       + c_4_163·a_1_22·b_1_32 + c_4_163·a_1_02·b_1_12 + c_4_163·a_1_23·b_1_3
       + c_4_163·a_1_03·b_1_1
  57. a_8_32·b_1_33·b_5_23 + a_8_32·b_8_43 + b_6_31·a_1_0·b_1_18·b_1_3
       + b_6_312·a_1_0·b_1_13 + b_4_152·b_6_31·a_1_0·b_1_1 + a_8_32·a_1_2·b_1_32·b_5_23
       + b_6_312·a_1_0·a_3_10 + a_8_32·a_1_22·b_1_3·b_5_23 + b_6_31·a_8_32·a_1_02
       + c_8_45·a_1_0·b_1_16·b_1_3 + c_8_45·a_1_0·b_1_17 + c_4_16·a_1_0·b_1_111
       + c_4_16·a_8_32·b_1_34 + c_4_16·a_8_32·b_1_14 + c_4_16·b_6_31·b_1_13·a_3_10
       + c_4_16·b_6_31·a_1_0·b_5_22 + c_4_16·b_6_31·a_1_0·b_1_15
       + b_4_15·c_8_45·a_1_0·b_1_13 + b_4_152·c_4_16·a_1_0·b_1_13
       + c_4_16·b_6_31·a_1_0·b_1_12·a_3_10 + c_8_45·a_1_02·b_1_13·a_3_10
       + c_4_16·a_8_32·a_1_22·b_1_32 + c_4_16·b_6_31·a_1_02·b_1_1·a_3_10
       + c_4_16·c_8_45·a_1_0·b_1_12·b_1_3 + c_4_16·c_8_45·a_1_0·b_1_13
       + c_4_162·b_1_15·a_3_10 + c_4_162·a_1_0·b_1_16·b_1_3
       + c_4_162·b_6_31·a_1_0·b_1_1 + b_4_15·c_4_162·a_1_0·b_1_13
       + c_4_162·a_1_0·b_1_14·a_3_10 + c_4_162·a_1_02·b_1_16
       + c_4_162·b_6_31·a_1_02 + c_4_16·c_8_45·a_1_03·b_1_1
       + c_4_162·a_1_02·b_1_13·a_3_10 + c_4_163·a_1_0·b_1_12·b_1_3
       + c_4_163·a_1_02·b_1_12 + c_4_163·a_1_03·b_1_1

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 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_4_16, a Duflot regular element of degree 4
    2. c_8_45, a Duflot regular element of degree 8
    3. b_1_32 + b_1_1·b_1_3 + b_1_12, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 11].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

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. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 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. a_3_11 → 0, an element of degree 3
  7. b_4_15 → 0, an element of degree 4
  8. c_4_16 → c_1_04, an element of degree 4
  9. b_5_22 → 0, an element of degree 5
  10. b_5_23 → 0, an element of degree 5
  11. b_6_31 → 0, an element of degree 6
  12. a_8_32 → 0, an element of degree 8
  13. b_8_43 → 0, an element of degree 8
  14. c_8_45 → 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_0 → 0, an element of degree 1
  2. a_1_2 → 0, an element of degree 1
  3. b_1_1 → 0, 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. a_3_11 → 0, an element of degree 3
  7. b_4_15 → 0, an element of degree 4
  8. c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. b_5_22 → 0, an element of degree 5
  10. b_5_23 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  11. b_6_31 → 0, an element of degree 6
  12. a_8_32 → 0, an element of degree 8
  13. b_8_43 → c_1_0·c_1_27 + c_1_04·c_1_24, an element of degree 8
  14. c_8_45 → c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_08, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 0, 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. a_3_11 → 0, an element of degree 3
  7. b_4_15 → 0, an element of degree 4
  8. c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. b_5_22 → c_1_25 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  10. b_5_23 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  11. b_6_31 → c_1_26 + c_1_0·c_1_25 + c_1_04·c_1_22, an element of degree 6
  12. a_8_32 → 0, an element of degree 8
  13. b_8_43 → 0, an element of degree 8
  14. c_8_45 → c_1_28 + c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_08, an element of degree 8

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

  1. a_1_0 → 0, an element of degree 1
  2. a_1_2 → 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. a_3_11 → 0, an element of degree 3
  7. b_4_15 → c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
  8. c_4_16 → c_1_0·c_1_23 + c_1_04, an element of degree 4
  9. b_5_22 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
  10. b_5_23 → c_1_25, an element of degree 5
  11. b_6_31 → c_1_26 + c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6
  12. a_8_32 → 0, an element of degree 8
  13. b_8_43 → 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_04·c_1_24, an element of degree 8
  14. c_8_45 → c_1_28 + 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_26 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_08, 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