Cohomology of group number 1825 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 4.
  • Its center has rank 2.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 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
    t3  −  t2  +  1
    (t  −  1)4 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-4,-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 14 minimal generators of maximal degree 8:

  1. a_1_2, a nilpotent element of degree 1
  2. b_1_0, an 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_4, a nilpotent element of degree 3
  6. b_3_10, an element of degree 3
  7. b_3_12, an element of degree 3
  8. c_4_19, a Duflot regular element of degree 4
  9. a_5_8, a nilpotent element of degree 5
  10. b_5_28, an element of degree 5
  11. b_6_38, an element of degree 6
  12. a_7_28, a nilpotent element of degree 7
  13. b_7_51, an element of degree 7
  14. c_8_68, 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 53 minimal relations of maximal degree 14:

  1. a_1_2·b_1_0
  2. b_1_0·b_1_1 + a_1_22
  3. a_1_22·b_1_1
  4. b_1_0·b_1_32 + a_1_2·b_1_12
  5. b_1_0·a_3_4
  6. b_1_1·b_3_10 + b_1_14 + a_1_2·a_3_4 + a_1_22·b_1_32
  7. a_1_2·b_3_10 + a_1_22·b_1_32
  8. b_1_0·b_3_12 + a_1_2·a_3_4 + a_1_22·b_1_32
  9. a_1_2·b_1_12·b_1_32
  10. b_1_32·b_3_10 + b_1_13·b_1_32 + a_1_2·b_1_1·b_3_12 + a_1_22·b_3_12
       + a_1_22·b_1_33
  11. a_3_4·b_3_10 + b_1_13·a_3_4
  12. b_3_10·b_3_12 + b_1_13·b_3_12 + a_3_42 + a_1_22·b_1_3·b_3_12
  13. b_3_102 + b_1_16 + c_4_19·b_1_02
  14. a_3_42 + c_4_19·a_1_22
  15. b_1_02·b_1_3·b_3_10 + b_1_0·a_5_8
  16. b_3_122 + b_1_12·b_1_34 + b_1_16 + a_1_2·b_1_1·b_1_34 + a_3_42 + a_1_2·a_5_8
       + a_1_22·b_1_3·b_3_12 + c_4_19·b_1_12
  17. b_3_122 + b_1_1·b_5_28 + b_1_1·b_1_32·b_3_12 + b_1_12·b_1_3·b_3_12 + b_1_13·b_3_12
       + b_1_13·b_1_33 + b_1_16 + b_1_1·a_5_8 + b_1_1·b_1_32·a_3_4 + a_1_2·b_1_32·b_3_12
       + a_1_2·b_1_1·b_1_3·b_3_12 + a_3_42 + a_1_22·b_1_3·b_3_12 + c_4_19·b_1_1·b_1_3
  18. b_3_122 + b_1_12·b_1_34 + b_1_16 + a_1_2·b_5_28 + a_1_2·b_1_32·b_3_12
       + a_1_2·b_1_1·b_1_3·b_3_12 + a_3_42 + a_1_2·b_1_32·a_3_4 + c_4_19·b_1_12
       + c_4_19·a_1_2·b_1_3 + c_4_19·a_1_2·b_1_1
  19. a_1_2·b_1_1·b_1_32·b_3_12 + a_1_2·b_1_1·a_5_8
  20. b_1_32·b_5_28 + b_1_34·b_3_12 + b_1_1·b_1_33·b_3_12 + b_1_1·b_1_36
       + b_1_12·b_1_32·b_3_12 + b_1_12·b_1_35 + b_1_32·a_5_8 + b_1_12·a_5_8
       + b_1_12·b_1_32·a_3_4 + b_1_13·b_1_3·a_3_4 + a_1_2·b_1_33·b_3_12
       + a_1_2·b_1_33·a_3_4 + c_4_19·b_1_33 + c_4_19·b_1_1·b_1_32
  21. b_1_02·b_5_28 + b_1_04·b_3_10 + b_1_06·b_1_3 + b_6_38·b_1_0 + a_1_2·b_1_33·a_3_4
       + c_4_19·b_1_03 + c_4_19·a_1_2·b_1_12
  22. b_1_34·a_3_4 + b_1_12·a_5_8 + b_1_12·b_1_32·a_3_4 + b_1_13·b_1_3·a_3_4
       + a_1_2·b_1_33·b_3_12 + a_1_2·b_1_1·b_1_32·b_3_12 + a_1_2·b_1_1·b_1_35
       + b_6_38·a_1_2 + a_1_2·b_1_3·a_5_8 + a_1_22·b_1_32·b_3_12 + c_4_19·a_1_2·b_1_32
       + c_4_19·a_1_2·b_1_12 + c_4_19·a_1_22·b_1_3
  23. b_3_10·a_5_8 + b_1_13·a_5_8 + c_4_19·b_1_03·b_1_3
  24. b_3_12·b_5_28 + b_1_1·b_1_34·b_3_12 + b_1_12·b_1_33·b_3_12 + b_1_12·b_1_36
       + b_1_13·b_1_35 + b_1_14·b_1_34 + b_1_16·b_1_32 + b_1_17·b_1_3 + b_1_18
       + b_3_12·a_5_8 + b_1_32·a_3_4·b_3_12 + a_1_2·b_1_1·b_1_36 + b_6_38·a_1_2·b_1_1
       + a_3_4·a_5_8 + a_1_2·b_1_32·a_5_8 + a_1_2·b_1_1·b_1_3·a_5_8 + c_4_19·b_1_3·b_3_12
       + c_4_19·b_1_1·b_3_12 + c_4_19·b_1_12·b_1_32 + c_4_19·b_1_13·b_1_3
       + c_4_19·b_1_14 + c_4_19·a_1_2·b_1_12·b_1_3
  25. b_3_12·b_5_28 + b_1_1·b_1_34·b_3_12 + b_1_12·b_1_33·b_3_12 + b_1_12·b_1_36
       + b_1_13·b_1_35 + b_1_14·b_1_34 + b_1_16·b_1_32 + b_1_17·b_1_3 + b_1_18
       + b_3_12·a_5_8 + a_3_4·b_5_28 + b_1_1·b_1_3·a_3_4·b_3_12 + b_1_12·a_3_4·b_3_12
       + b_1_12·b_1_33·a_3_4 + b_1_13·a_5_8 + b_1_13·b_1_32·a_3_4 + b_1_14·b_1_3·a_3_4
       + a_1_2·b_1_1·b_1_36 + a_1_2·b_1_32·a_5_8 + a_1_2·b_1_1·b_1_3·a_5_8
       + b_6_38·a_1_22 + c_4_19·b_1_3·b_3_12 + c_4_19·b_1_1·b_3_12 + c_4_19·b_1_12·b_1_32
       + c_4_19·b_1_13·b_1_3 + c_4_19·b_1_14 + c_4_19·b_1_3·a_3_4 + c_4_19·b_1_1·a_3_4
       + c_4_19·a_1_2·b_1_1·b_1_32 + c_4_19·a_1_2·b_1_12·b_1_3
  26. b_1_07·b_1_3 + b_6_38·b_1_0·b_1_3 + b_1_0·a_7_28 + b_1_03·a_5_8 + c_4_19·b_1_03·b_1_3
       + c_4_19·a_1_2·b_1_12·b_1_3
  27. a_3_4·a_5_8 + a_1_2·a_7_28 + a_1_2·b_1_1·b_1_3·a_5_8
  28. b_3_12·b_5_28 + b_1_1·b_7_51 + b_1_13·b_1_35 + b_1_14·b_1_34 + b_1_16·b_1_32
       + b_1_18 + b_6_38·b_1_1·b_1_3 + b_3_12·a_5_8 + a_3_4·b_5_28 + b_1_32·a_3_4·b_3_12
       + b_1_1·b_1_32·a_5_8 + b_1_12·b_1_33·a_3_4 + b_1_13·a_5_8 + b_1_13·b_1_32·a_3_4
       + b_1_15·a_3_4 + a_1_2·b_1_1·b_1_36 + a_3_4·a_5_8 + a_1_2·b_1_32·a_5_8
       + a_1_2·b_1_1·b_1_3·a_5_8 + c_4_19·b_1_3·b_3_12 + c_4_19·b_1_1·b_1_33
       + c_4_19·b_1_14 + c_4_19·b_1_3·a_3_4 + c_4_19·a_1_2·b_1_12·b_1_3
       + c_4_19·a_1_22·b_1_32
  29. b_3_10·b_5_28 + b_1_13·b_1_32·b_3_12 + b_1_14·b_1_3·b_3_12 + b_1_14·b_1_34
       + b_1_15·b_3_12 + b_1_15·b_1_33 + b_1_0·b_7_51 + b_6_38·b_1_02 + b_1_13·a_5_8
       + b_1_13·b_1_32·a_3_4 + c_4_19·b_1_3·b_3_10 + c_4_19·b_1_14 + c_4_19·b_1_04
  30. b_3_12·b_5_28 + b_1_1·b_1_34·b_3_12 + b_1_12·b_1_33·b_3_12 + b_1_12·b_1_36
       + b_1_13·b_1_35 + b_1_14·b_1_34 + b_1_16·b_1_32 + b_1_17·b_1_3 + b_1_18
       + b_3_12·a_5_8 + b_1_32·a_3_4·b_3_12 + a_1_2·b_7_51 + a_1_2·b_1_1·b_1_36
       + b_6_38·a_1_2·b_1_3 + a_3_4·a_5_8 + a_1_2·b_1_32·a_5_8 + c_4_19·b_1_3·b_3_12
       + c_4_19·b_1_1·b_3_12 + c_4_19·b_1_12·b_1_32 + c_4_19·b_1_13·b_1_3
       + c_4_19·b_1_14 + c_4_19·a_1_2·b_3_12 + c_4_19·a_1_2·b_1_33 + c_4_19·a_1_2·a_3_4
       + c_4_19·a_1_22·b_1_32
  31. b_1_33·a_3_4·b_3_12 + b_1_12·a_7_28 + b_1_12·b_1_3·a_3_4·b_3_12
       + b_1_13·a_3_4·b_3_12 + b_1_14·a_5_8 + b_6_38·a_3_4 + b_6_38·a_1_2·b_1_1·b_1_3
       + a_1_2·b_1_3·a_7_28 + a_1_2·b_1_1·a_7_28 + a_1_2·b_1_1·b_1_32·a_5_8
       + c_4_19·b_1_32·a_3_4 + c_4_19·b_1_12·a_3_4 + c_4_19·a_1_2·b_1_34
       + c_4_19·a_1_2·b_1_1·b_1_33 + c_4_19·a_1_2·b_1_3·a_3_4 + c_4_19·a_1_22·b_3_12
  32. b_1_32·b_7_51 + b_1_36·b_3_12 + b_1_1·b_1_35·b_3_12 + b_1_1·b_1_38
       + b_1_16·b_1_33 + b_6_38·b_1_33 + b_1_33·a_3_4·b_3_12 + b_1_34·a_5_8
       + b_1_1·b_3_12·a_5_8 + b_1_12·b_1_3·a_3_4·b_3_12 + b_1_13·b_1_33·a_3_4
       + b_1_14·b_1_32·a_3_4 + b_6_38·a_1_2·b_1_32 + b_6_38·a_1_2·b_1_1·b_1_3
       + a_1_2·b_1_1·a_7_28 + c_4_19·b_1_32·b_3_12 + c_4_19·b_1_35 + c_4_19·b_1_1·b_1_34
       + c_4_19·b_1_12·b_1_33 + c_4_19·b_1_32·a_3_4 + c_4_19·a_1_2·b_1_34
       + c_4_19·a_1_2·b_1_1·b_1_33 + c_4_19·a_1_22·b_1_33
  33. b_1_02·b_7_51 + b_6_38·b_3_10 + b_6_38·b_1_13 + b_6_38·b_1_03 + b_1_04·a_5_8
       + a_1_2·b_1_1·a_7_28 + c_4_19·b_1_0·b_1_3·b_3_10 + c_4_19·b_1_02·b_3_10
       + c_4_19·a_1_2·b_1_1·b_3_12
  34. b_6_38·b_1_3·b_3_10 + b_6_38·b_1_13·b_1_3 + a_5_8·b_5_28 + b_1_32·b_3_12·a_5_8
       + b_1_1·b_1_3·b_3_12·a_5_8 + b_1_1·b_1_34·a_5_8 + b_1_12·b_1_3·a_7_28
       + b_1_12·b_1_33·a_5_8 + b_1_14·b_1_3·a_5_8 + b_6_38·b_1_3·a_3_4 + a_5_82
       + a_1_2·b_1_1·b_1_3·a_7_28 + a_1_2·b_1_1·b_1_33·a_5_8 + c_4_19·b_1_05·b_1_3
       + c_4_19·b_1_3·a_5_8 + c_4_19·b_1_33·a_3_4 + c_4_19·b_1_1·a_5_8
       + c_4_19·b_1_12·b_1_3·a_3_4 + c_4_19·b_1_0·a_5_8 + c_4_19·a_1_2·b_1_35
       + c_4_19·a_1_2·b_1_1·b_1_3·b_3_12 + c_4_19·a_1_2·b_1_1·b_1_34
       + c_4_19·a_1_2·b_1_32·a_3_4 + c_4_19·a_1_22·b_1_3·b_3_12
  35. a_5_8·b_5_28 + b_3_10·a_7_28 + b_1_32·b_3_12·a_5_8 + b_1_1·b_1_3·b_3_12·a_5_8
       + b_1_1·b_1_34·a_5_8 + b_1_12·b_1_3·a_7_28 + b_1_12·b_1_33·a_5_8 + b_1_13·a_7_28
       + b_1_14·b_1_3·a_5_8 + b_1_05·a_5_8 + b_6_38·b_1_3·a_3_4 + a_5_82
       + a_1_2·b_1_1·b_1_33·a_5_8 + c_4_19·b_1_3·a_5_8 + c_4_19·b_1_33·a_3_4
       + c_4_19·b_1_1·a_5_8 + c_4_19·b_1_12·b_1_3·a_3_4 + c_4_19·a_1_2·b_1_35
       + c_4_19·a_1_2·b_1_1·b_1_34 + c_4_19·a_1_2·b_1_32·a_3_4
       + c_4_19·a_1_22·b_1_3·b_3_12
  36. b_6_38·b_1_3·b_3_10 + b_6_38·b_1_13·b_1_3 + a_5_8·b_5_28 + b_1_32·b_3_12·a_5_8
       + b_1_1·b_1_3·b_3_12·a_5_8 + b_1_1·b_1_34·a_5_8 + b_1_12·b_1_3·a_7_28
       + b_1_12·b_1_33·a_5_8 + b_1_14·b_1_3·a_5_8 + b_6_38·b_1_3·a_3_4 + a_5_82
       + a_3_4·a_7_28 + a_1_2·b_1_1·b_1_33·a_5_8 + c_4_19·b_1_05·b_1_3 + c_4_19·b_1_3·a_5_8
       + c_4_19·b_1_33·a_3_4 + c_4_19·b_1_1·a_5_8 + c_4_19·b_1_12·b_1_3·a_3_4
       + c_4_19·b_1_0·a_5_8 + c_4_19·a_1_2·b_1_35 + c_4_19·a_1_2·b_1_1·b_1_3·b_3_12
       + c_4_19·a_1_2·b_1_1·b_1_34 + c_4_19·a_1_2·a_5_8 + c_4_19·a_1_2·b_1_32·a_3_4
  37. b_3_12·b_7_51 + b_1_1·b_1_36·b_3_12 + b_1_12·b_1_38 + b_1_13·b_1_37
       + b_1_16·b_1_3·b_3_12 + b_1_16·b_1_34 + b_1_17·b_1_33 + b_6_38·b_1_3·b_3_12
       + b_1_32·b_3_12·a_5_8 + b_1_13·b_1_3·a_3_4·b_3_12 + b_1_13·b_1_32·a_5_8
       + b_1_14·a_3_4·b_3_12 + b_1_14·b_1_3·a_5_8 + b_1_16·b_1_3·a_3_4 + b_1_17·a_3_4
       + a_1_2·b_1_1·b_1_38 + b_6_38·a_1_2·b_1_1·b_1_32 + a_1_2·b_1_32·a_7_28
       + a_1_2·b_1_34·a_5_8 + c_4_19·b_1_33·b_3_12 + c_4_19·b_1_1·b_1_32·b_3_12
       + c_4_19·b_1_12·b_1_3·b_3_12 + c_4_19·b_1_13·b_1_33 + c_4_19·b_1_16
       + c_4_19·a_3_4·b_3_12 + c_4_19·b_1_1·b_1_32·a_3_4 + c_4_19·b_1_12·b_1_3·a_3_4
       + c_4_19·b_1_13·a_3_4 + c_4_19·a_1_2·b_1_32·b_3_12 + c_4_19·a_1_22·b_1_3·b_3_12
       + c_4_192·b_1_12 + c_4_192·a_1_22
  38. b_3_10·b_7_51 + b_1_13·b_1_34·b_3_12 + b_1_14·b_1_33·b_3_12 + b_1_14·b_1_36
       + b_1_19·b_1_3 + b_6_38·b_1_3·b_3_10 + b_6_38·b_1_0·b_3_10 + a_5_8·b_5_28
       + b_1_32·b_3_12·a_5_8 + b_1_1·b_1_3·b_3_12·a_5_8 + b_1_1·b_1_34·a_5_8
       + b_1_12·b_1_3·a_7_28 + b_1_12·b_1_32·a_3_4·b_3_12 + b_1_12·b_1_33·a_5_8
       + b_1_13·b_1_3·a_3_4·b_3_12 + b_1_13·b_1_32·a_5_8 + b_1_14·a_3_4·b_3_12
       + b_1_14·b_1_3·a_5_8 + b_1_16·b_1_3·a_3_4 + b_1_17·a_3_4 + b_6_38·b_1_3·a_3_4
       + a_5_82 + a_1_2·b_1_1·b_1_33·a_5_8 + c_4_19·b_1_13·b_3_12
       + c_4_19·b_1_13·b_1_33 + c_4_19·b_1_14·b_1_32 + c_4_19·b_1_15·b_1_3
       + c_4_19·b_1_0·b_5_28 + c_4_19·b_1_03·b_3_10 + c_4_19·b_1_05·b_1_3
       + c_4_19·b_1_3·a_5_8 + c_4_19·b_1_33·a_3_4 + c_4_19·b_1_1·a_5_8
       + c_4_19·b_1_12·b_1_3·a_3_4 + c_4_19·b_1_13·a_3_4 + c_4_19·b_1_0·a_5_8
       + c_4_19·a_1_2·b_1_35 + c_4_19·a_1_2·b_1_1·b_1_3·b_3_12
       + c_4_19·a_1_2·b_1_1·b_1_34 + c_4_19·a_1_2·b_1_32·a_3_4
       + c_4_19·a_1_22·b_1_3·b_3_12 + c_4_192·b_1_0·b_1_3
  39. a_3_4·b_7_51 + b_1_12·b_1_3·a_7_28 + b_1_12·b_1_32·a_3_4·b_3_12 + b_1_13·a_7_28
       + b_1_13·b_1_32·a_5_8 + b_1_14·a_3_4·b_3_12 + b_1_14·b_1_3·a_5_8
       + b_1_14·b_1_33·a_3_4 + b_1_15·b_1_32·a_3_4 + b_6_38·b_1_1·a_3_4
       + c_4_19·a_3_4·b_3_12 + c_4_19·b_1_13·a_3_4 + c_4_19·a_1_2·b_1_35
       + c_4_19·a_1_2·b_1_1·b_1_34 + c_4_19·a_1_2·b_1_32·a_3_4
       + c_4_19·a_1_22·b_1_3·b_3_12 + c_4_192·a_1_22
  40. b_5_282 + b_1_18·b_1_32 + b_1_110 + b_1_03·a_7_28 + b_1_05·a_5_8
       + a_1_2·b_1_1·b_1_38 + a_5_82 + a_1_2·b_1_34·a_5_8 + c_8_68·b_1_02
       + c_4_19·b_1_12·b_1_34 + c_4_19·b_1_14·b_1_32 + c_4_19·b_1_16
       + c_4_192·b_1_32 + c_4_192·b_1_12
  41. a_1_2·b_1_1·b_1_38 + a_5_82 + a_1_2·b_1_34·a_5_8 + c_8_68·a_1_22
  42. b_6_38·b_5_28 + b_6_38·b_1_32·b_3_12 + b_6_38·b_1_1·b_1_3·b_3_12
       + b_6_38·b_1_1·b_1_34 + b_6_38·b_1_12·b_3_12 + b_6_38·b_1_12·b_1_33
       + b_6_38·b_1_02·b_3_10 + b_6_38·b_1_04·b_1_3 + b_1_0·b_3_10·a_7_28 + b_1_04·a_7_28
       + b_6_38·a_5_8 + b_6_38·b_1_32·a_3_4 + b_6_38·a_1_2·b_1_1·b_1_33
       + a_1_2·b_1_33·a_7_28 + a_1_2·b_1_1·b_1_34·a_5_8 + c_8_68·b_1_03
       + c_4_19·b_1_0·b_1_3·b_5_28 + c_4_19·b_1_06·b_1_3 + c_4_19·b_1_07
       + c_4_19·b_6_38·b_1_3 + c_4_19·b_6_38·b_1_1 + c_4_19·b_6_38·b_1_0
       + c_4_19·a_1_2·b_1_36 + c_4_19·b_6_38·a_1_2 + c_4_19·a_1_2·b_1_3·a_5_8
       + c_4_19·a_1_2·b_1_33·a_3_4 + c_4_19·a_1_2·b_1_1·a_5_8 + c_4_192·b_1_02·b_1_3
       + c_4_192·b_1_03 + c_4_192·a_1_2·b_1_32 + c_4_192·a_1_2·b_1_12
  43. b_1_33·b_3_12·a_5_8 + b_1_34·a_7_28 + b_1_12·b_1_34·a_5_8 + b_1_0·b_3_10·a_7_28
       + b_1_06·a_5_8 + b_6_38·a_5_8 + b_6_38·b_1_32·a_3_4 + b_6_38·b_1_1·b_1_3·a_3_4
       + b_6_38·a_1_2·b_1_34 + b_6_38·a_1_2·b_1_1·b_1_33 + b_1_3·a_5_82
       + a_1_2·b_1_33·a_7_28 + a_1_2·b_1_1·b_1_32·a_7_28 + c_4_19·b_1_06·b_1_3
       + c_8_68·a_1_2·b_1_12 + c_4_19·b_1_32·a_5_8 + c_4_19·b_1_1·b_1_33·a_3_4
       + c_4_19·b_1_02·a_5_8 + c_4_19·a_1_2·b_1_33·b_3_12 + c_4_19·a_1_2·b_1_1·b_1_35
       + c_4_19·b_6_38·a_1_2 + c_4_19·a_1_2·b_1_33·a_3_4 + c_4_19·a_1_22·b_1_32·b_3_12
       + c_4_192·a_1_2·b_1_32 + c_4_192·a_1_2·b_1_12 + c_4_192·a_1_22·b_1_3
  44. b_6_38·b_1_05·b_1_3 + b_5_28·a_7_28 + b_1_32·b_3_12·a_7_28
       + b_1_1·b_1_3·b_3_12·a_7_28 + b_1_1·b_1_34·a_7_28 + b_1_12·b_1_33·a_7_28
       + b_1_14·b_1_3·a_7_28 + b_1_14·b_1_33·a_5_8 + b_1_16·b_1_3·a_5_8
       + b_1_16·b_1_33·a_3_4 + b_1_18·b_1_3·a_3_4 + b_1_19·a_3_4 + b_1_07·a_5_8
       + b_6_38·a_3_4·b_3_12 + b_6_38·b_1_12·b_1_3·a_3_4 + a_5_8·a_7_28
       + a_1_2·b_1_1·b_1_33·a_7_28 + c_8_68·b_1_03·b_1_3 + c_4_19·b_6_38·b_1_0·b_1_3
       + c_4_19·b_1_3·a_7_28 + c_4_19·b_1_32·a_3_4·b_3_12 + c_4_19·b_1_1·a_7_28
       + c_4_19·b_1_12·a_3_4·b_3_12 + c_4_19·b_1_15·a_3_4 + c_4_19·b_1_0·a_7_28
       + c_4_19·b_1_03·a_5_8 + c_4_19·a_1_2·b_1_1·b_1_36 + c_4_19·b_6_38·a_1_2·b_1_1
       + c_4_19·a_1_2·b_1_32·a_5_8 + c_4_19·b_6_38·a_1_22 + c_4_192·b_1_03·b_1_3
       + c_4_192·a_1_2·b_1_1·b_1_32 + c_4_192·a_1_2·b_1_12·b_1_3
  45. a_5_8·b_7_51 + b_1_35·a_7_28 + b_1_1·b_1_34·a_7_28 + b_1_1·b_1_36·a_5_8
       + b_1_12·b_1_35·a_5_8 + b_1_13·b_1_34·a_5_8 + b_1_16·b_1_3·a_5_8
       + b_6_38·b_1_33·a_3_4 + b_6_38·b_1_1·a_5_8 + b_6_38·b_1_12·b_1_3·a_3_4
       + b_6_38·b_1_0·a_5_8 + b_6_38·a_1_2·b_1_35 + a_1_2·b_1_1·b_1_33·a_7_28
       + c_4_19·b_6_38·b_1_0·b_1_3 + c_8_68·a_1_2·b_1_12·b_1_3 + c_4_19·b_3_12·a_5_8
       + c_4_19·b_1_12·b_1_3·a_5_8 + c_4_19·b_1_12·b_1_33·a_3_4 + c_4_19·b_1_13·a_5_8
       + c_4_19·b_1_13·b_1_32·a_3_4 + c_4_19·b_1_14·b_1_3·a_3_4
       + c_4_19·b_6_38·a_1_2·b_1_3 + c_4_19·b_6_38·a_1_2·b_1_1 + c_8_68·a_1_22·b_1_32
       + c_4_19·a_1_2·a_7_28 + c_4_19·a_1_2·b_1_32·a_5_8 + c_4_19·a_1_2·b_1_1·b_1_3·a_5_8
       + c_4_19·b_6_38·a_1_22 + c_4_192·b_1_03·b_1_3 + c_4_192·a_1_2·b_1_33
       + c_4_192·a_1_2·b_1_1·b_1_32 + c_4_192·a_1_22·b_1_32
  46. b_1_12·b_1_37·b_3_12 + b_1_13·b_1_39 + b_1_15·b_1_34·b_3_12 + b_1_15·b_1_37
       + b_1_16·b_1_36 + b_1_17·b_1_35 + b_1_18·b_1_3·b_3_12 + b_1_18·b_1_34
       + b_1_19·b_3_12 + b_1_110·b_1_32 + b_1_112 + b_6_38·b_1_12·b_1_34
       + b_6_38·b_1_15·b_1_3 + b_6_382 + b_1_12·b_1_35·a_5_8 + b_1_13·b_1_34·a_5_8
       + b_1_14·b_1_3·a_7_28 + b_1_15·a_7_28 + b_1_15·b_1_32·a_5_8 + b_1_16·b_1_33·a_3_4
       + b_1_17·a_5_8 + b_1_17·b_1_32·a_3_4 + b_1_19·a_3_4 + b_1_05·a_7_28 + b_1_07·a_5_8
       + b_6_38·b_1_12·b_1_3·a_3_4 + c_8_68·b_1_14 + c_8_68·b_1_04 + c_4_19·b_1_38
       + c_4_19·b_1_15·b_1_33 + c_4_19·b_1_16·b_1_32 + c_4_19·b_1_18 + c_4_19·b_1_08
       + c_4_19·b_1_12·b_1_33·a_3_4 + c_4_19·b_1_14·b_1_3·a_3_4 + c_8_68·a_1_22·b_1_32
       + c_4_192·b_1_34 + c_4_192·b_1_14 + c_4_192·b_1_04 + c_4_192·a_1_22·b_1_32
  47. b_5_28·b_7_51 + b_1_12·b_1_37·b_3_12 + b_1_13·b_1_39 + b_1_15·b_1_37
       + b_1_16·b_1_33·b_3_12 + b_1_16·b_1_36 + b_1_17·b_1_32·b_3_12 + b_1_17·b_1_35
       + b_1_18·b_1_3·b_3_12 + b_1_18·b_1_34 + b_1_19·b_1_33 + b_6_38·b_1_33·b_3_12
       + b_6_38·b_1_1·b_1_32·b_3_12 + b_6_38·b_1_1·b_1_35 + b_6_38·b_1_12·b_1_3·b_3_12
       + b_6_38·b_1_12·b_1_34 + b_6_38·b_1_03·b_3_10 + b_6_38·b_1_05·b_1_3
       + b_1_12·b_1_33·a_7_28 + b_1_12·b_1_35·a_5_8 + b_1_13·b_1_32·a_7_28
       + b_1_14·b_1_32·a_3_4·b_3_12 + b_1_14·b_1_33·a_5_8 + b_1_15·a_7_28
       + b_1_15·b_1_3·a_3_4·b_3_12 + b_1_17·b_1_32·a_3_4 + b_1_18·b_1_3·a_3_4
       + b_1_19·a_3_4 + b_1_05·a_7_28 + b_6_38·b_1_3·a_5_8 + b_6_38·b_1_1·b_1_32·a_3_4
       + b_6_38·b_1_13·a_3_4 + b_6_38·b_1_0·a_5_8 + b_6_38·a_1_2·b_1_1·b_1_34
       + b_6_38·a_1_2·a_5_8 + c_8_68·b_1_0·b_3_10 + c_8_68·b_1_04 + c_4_19·b_1_3·b_7_51
       + c_4_19·b_1_35·b_3_12 + c_4_19·b_1_1·b_1_37 + c_4_19·b_1_12·b_1_33·b_3_12
       + c_4_19·b_1_12·b_1_36 + c_4_19·b_1_13·b_1_35 + c_4_19·b_1_14·b_1_3·b_3_12
       + c_4_19·b_1_15·b_1_33 + c_4_19·b_1_16·b_1_32 + c_4_19·b_1_18
       + c_4_19·b_1_05·b_3_10 + c_4_19·b_1_08 + c_4_19·b_6_38·b_1_1·b_1_3
       + c_4_19·b_3_12·a_5_8 + c_4_19·b_1_32·a_3_4·b_3_12 + c_4_19·b_1_33·a_5_8
       + c_4_19·b_1_12·b_1_3·a_5_8 + c_4_19·b_1_12·b_1_33·a_3_4
       + c_4_19·b_1_13·b_1_32·a_3_4 + c_4_19·b_1_14·b_1_3·a_3_4 + c_4_19·b_1_15·a_3_4
       + c_4_19·b_1_0·a_7_28 + c_4_19·a_1_2·b_1_1·b_1_36 + c_4_19·b_6_38·a_1_2·b_1_1
       + c_8_68·a_1_22·b_1_32 + c_4_19·a_1_2·a_7_28 + c_4_192·b_1_1·b_3_12
       + c_4_192·b_1_1·b_1_33 + c_4_192·b_1_14 + c_4_192·b_1_1·a_3_4
       + c_4_192·a_1_2·b_1_1·b_1_32 + c_4_192·a_1_2·b_1_12·b_1_3
  48. b_6_38·a_1_2·b_1_1·b_1_34 + a_5_8·a_7_28 + a_1_2·b_1_1·b_1_35·a_5_8
       + c_4_19·a_1_2·b_1_1·b_1_36 + c_8_68·a_1_2·a_3_4 + c_4_19·a_1_2·b_1_1·b_1_3·a_5_8
  49. b_6_38·b_7_51 + b_6_38·b_1_34·b_3_12 + b_6_38·b_1_1·b_1_33·b_3_12
       + b_6_38·b_1_1·b_1_36 + b_6_38·b_1_16·b_1_3 + b_6_382·b_1_3
       + b_1_1·b_1_32·b_3_12·a_7_28 + b_1_13·b_1_33·a_7_28 + b_1_13·b_1_35·a_5_8
       + b_1_15·b_1_33·a_5_8 + b_1_17·b_1_3·a_5_8 + b_1_06·a_7_28
       + b_6_38·b_1_3·a_3_4·b_3_12 + b_6_38·b_1_32·a_5_8 + b_6_38·b_1_1·a_3_4·b_3_12
       + b_6_38·b_1_1·b_1_33·a_3_4 + b_6_38·b_1_13·b_1_3·a_3_4 + b_6_38·b_1_14·a_3_4
       + b_6_38·a_1_2·b_1_1·a_5_8 + c_8_68·b_1_02·b_3_10 + c_8_68·b_1_04·b_1_3
       + c_8_68·b_1_05 + c_4_19·b_1_0·b_1_3·b_7_51 + c_4_19·b_1_06·b_3_10 + c_4_19·b_1_09
       + c_4_19·b_6_38·b_3_12 + c_4_19·b_6_38·b_3_10 + c_4_19·b_6_38·b_1_33
       + c_4_19·b_6_38·b_1_1·b_1_32 + c_4_19·b_6_38·b_1_12·b_1_3 + c_4_19·b_6_38·b_1_13
       + c_4_19·b_1_1·b_3_12·a_5_8 + c_4_19·b_1_12·b_1_3·a_3_4·b_3_12
       + c_4_19·b_1_14·b_1_32·a_3_4 + c_4_19·b_1_02·a_7_28 + c_4_19·a_1_2·b_1_38
       + c_4_19·b_6_38·a_3_4 + c_4_19·b_6_38·a_1_2·b_1_32 + c_8_68·a_1_22·b_3_12
       + c_4_19·a_1_2·b_1_1·a_7_28 + c_4_19·a_1_2·b_1_1·b_1_32·a_5_8
       + c_4_192·b_1_0·b_1_3·b_3_10 + c_4_192·b_1_02·b_3_10 + c_4_192·b_1_05
       + c_4_192·a_1_2·b_1_34 + c_4_192·a_1_2·b_1_1·b_3_12 + c_4_192·a_1_2·b_1_3·a_3_4
  50. b_1_33·b_3_12·a_7_28 + b_1_12·b_1_36·a_5_8 + b_1_13·b_1_35·a_5_8
       + b_1_14·b_1_32·a_7_28 + b_1_15·b_1_32·a_3_4·b_3_12 + b_1_16·a_7_28
       + b_1_17·b_1_3·a_5_8 + b_1_17·b_1_33·a_3_4 + b_1_18·b_1_32·a_3_4
       + b_1_19·b_1_3·a_3_4 + b_1_110·a_3_4 + b_1_0·b_5_28·a_7_28 + b_1_08·a_5_8
       + b_6_38·a_7_28 + b_6_38·b_1_3·a_3_4·b_3_12 + b_6_38·b_1_1·a_3_4·b_3_12
       + b_6_38·b_1_14·a_3_4 + b_6_38·b_1_02·a_5_8 + a_1_2·b_1_1·b_1_36·a_5_8
       + c_4_19·b_6_38·b_1_02·b_1_3 + c_8_68·b_1_12·a_3_4 + c_4_19·b_1_32·a_7_28
       + c_4_19·b_1_34·a_5_8 + c_4_19·b_1_1·b_1_32·a_3_4·b_3_12
       + c_4_19·b_1_12·b_1_32·a_5_8 + c_4_19·b_1_13·b_1_3·a_5_8
       + c_4_19·b_1_13·b_1_33·a_3_4 + c_4_19·a_1_2·b_1_38 + c_4_19·a_1_2·b_1_1·b_1_37
       + c_4_19·b_6_38·a_3_4 + c_4_19·b_6_38·a_1_2·b_1_1·b_1_3 + c_8_68·a_1_2·b_1_3·a_3_4
       + c_8_68·a_1_22·b_3_12 + c_8_68·a_1_22·b_1_33 + c_4_192·b_1_04·b_1_3
       + c_4_192·b_1_32·a_3_4 + c_4_192·b_1_12·a_3_4 + c_4_192·a_1_2·b_1_34
       + c_4_192·a_1_2·b_1_1·b_1_33 + c_4_192·a_1_2·b_1_3·a_3_4
       + c_4_192·a_1_22·b_3_12
  51. a_7_282 + c_4_19·a_5_82
  52. b_7_512 + b_1_14·b_1_310 + b_1_16·b_1_38 + b_1_18·b_1_36 + b_1_112·b_1_32
       + b_6_382·b_1_32 + b_1_07·a_7_28 + b_1_09·a_5_8 + c_8_68·b_1_06
       + c_4_19·b_1_12·b_1_38 + c_4_19·b_1_14·b_1_36 + c_4_19·b_1_010
       + c_4_19·b_1_03·a_7_28 + c_4_19·b_1_05·a_5_8 + c_4_19·c_8_68·b_1_02
       + c_4_192·b_1_36 + c_4_192·b_1_14·b_1_32 + c_4_192·b_1_16
       + c_4_192·a_1_2·b_1_1·b_1_34 + c_4_192·a_1_2·a_5_8
       + c_4_192·a_1_22·b_1_3·b_3_12 + c_4_193·b_1_12
  53. a_7_28·b_7_51 + b_1_1·b_1_36·a_7_28 + b_1_12·b_1_37·a_5_8 + b_1_14·b_1_33·a_7_28
       + b_1_14·b_1_35·a_5_8 + b_1_15·b_1_32·a_7_28 + b_1_16·b_1_32·a_3_4·b_3_12
       + b_1_17·b_1_3·a_3_4·b_3_12 + b_1_17·b_1_32·a_5_8 + b_1_18·a_3_4·b_3_12
       + b_1_18·b_1_3·a_5_8 + b_1_19·b_1_32·a_3_4 + b_1_110·b_1_3·a_3_4 + b_1_111·a_3_4
       + b_6_38·b_1_32·a_3_4·b_3_12 + b_6_38·b_1_1·a_7_28 + b_6_38·b_1_12·a_3_4·b_3_12
       + b_6_38·b_1_14·b_1_3·a_3_4 + b_6_38·b_1_0·a_7_28 + b_6_38·b_1_03·a_5_8
       + b_1_32·a_5_8·a_7_28 + a_1_2·b_1_1·b_1_37·a_5_8 + b_6_38·a_1_2·b_1_1·b_1_3·a_5_8
       + c_4_19·b_6_38·b_1_03·b_1_3 + c_8_68·b_1_12·b_1_3·a_3_4 + c_8_68·b_1_13·a_3_4
       + c_8_68·b_1_0·a_5_8 + c_4_19·b_3_12·a_7_28 + c_4_19·b_1_35·a_5_8
       + c_4_19·b_1_1·b_1_34·a_5_8 + c_4_19·b_1_12·b_1_3·a_7_28
       + c_4_19·b_1_12·b_1_32·a_3_4·b_3_12 + c_4_19·b_1_12·b_1_33·a_5_8
       + c_4_19·b_1_13·a_7_28 + c_4_19·b_1_13·b_1_3·a_3_4·b_3_12
       + c_4_19·b_1_14·a_3_4·b_3_12 + c_4_19·b_1_14·b_1_3·a_5_8
       + c_4_19·b_1_14·b_1_33·a_3_4 + c_4_19·b_1_16·b_1_3·a_3_4 + c_4_19·b_1_17·a_3_4
       + c_4_19·a_1_2·b_1_39 + c_4_19·b_6_38·b_1_3·a_3_4
       + c_4_19·b_6_38·a_1_2·b_1_1·b_1_32 + c_8_68·a_1_2·b_1_32·a_3_4
       + c_4_19·a_1_2·b_1_34·a_5_8 + c_4_19·a_1_2·b_1_1·b_1_3·a_7_28
       + c_4_192·b_1_05·b_1_3 + c_4_192·b_1_33·a_3_4 + c_4_192·b_1_12·b_1_3·a_3_4
       + c_4_192·a_1_2·b_1_35 + c_4_192·a_1_2·b_1_1·b_1_34 + c_4_192·a_1_2·a_5_8
       + c_4_192·a_1_2·b_1_32·a_3_4

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 14.
  • 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_8_68, a Duflot regular element of degree 8
    3. b_1_32 + b_1_1·b_1_3 + b_1_12 + b_1_02, an element of degree 2
    4. b_1_32, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 10, 12].
  • 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. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 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_4 → 0, an element of degree 3
  6. b_3_10 → 0, an element of degree 3
  7. b_3_12 → 0, an element of degree 3
  8. c_4_19 → c_1_04, an element of degree 4
  9. a_5_8 → 0, an element of degree 5
  10. b_5_28 → 0, an element of degree 5
  11. b_6_38 → 0, an element of degree 6
  12. a_7_28 → 0, an element of degree 7
  13. b_7_51 → 0, an element of degree 7
  14. c_8_68 → 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_2 → 0, an element of degree 1
  2. b_1_0 → c_1_2, 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_4 → 0, an element of degree 3
  6. b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
  7. b_3_12 → 0, an element of degree 3
  8. c_4_19 → c_1_02·c_1_22 + c_1_04, an element of degree 4
  9. a_5_8 → 0, an element of degree 5
  10. b_5_28 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
  11. b_6_38 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_25 + c_1_02·c_1_24, an element of degree 6
  12. a_7_28 → 0, an element of degree 7
  13. b_7_51 → c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_0·c_1_26 + c_1_0·c_1_12·c_1_24
       + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_12·c_1_23 + c_1_02·c_1_14·c_1_2
       + c_1_03·c_1_24 + c_1_05·c_1_22 + c_1_06·c_1_2, an element of degree 7
  14. c_8_68 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8

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

  1. a_1_2 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_1_1 → c_1_3, an element of degree 1
  4. b_1_3 → c_1_2, an element of degree 1
  5. a_3_4 → 0, an element of degree 3
  6. b_3_10 → c_1_33, an element of degree 3
  7. b_3_12 → c_1_33 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  8. c_4_19 → c_1_02·c_1_32 + c_1_04, an element of degree 4
  9. a_5_8 → 0, an element of degree 5
  10. b_5_28 → c_1_35 + c_1_2·c_1_34 + 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_22·c_1_3 + c_1_04·c_1_3 + c_1_04·c_1_2, an element of degree 5
  11. b_6_38 → c_1_24·c_1_32 + c_1_12·c_1_34 + c_1_14·c_1_32 + c_1_0·c_1_23·c_1_32
       + c_1_02·c_1_23·c_1_3 + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6
  12. a_7_28 → 0, an element of degree 7
  13. b_7_51 → c_1_2·c_1_36 + c_1_23·c_1_34 + c_1_24·c_1_33 + c_1_12·c_1_2·c_1_34
       + c_1_14·c_1_2·c_1_32 + c_1_0·c_1_23·c_1_33 + c_1_02·c_1_35
       + c_1_02·c_1_2·c_1_34 + c_1_02·c_1_25 + c_1_03·c_1_34 + c_1_04·c_1_2·c_1_32
       + c_1_05·c_1_32 + c_1_06·c_1_3, an element of degree 7
  14. c_8_68 → c_1_2·c_1_37 + c_1_23·c_1_35 + c_1_12·c_1_2·c_1_35 + c_1_12·c_1_24·c_1_32
       + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_24 + c_1_18
       + c_1_0·c_1_37 + c_1_0·c_1_2·c_1_36 + c_1_02·c_1_2·c_1_35
       + c_1_02·c_1_22·c_1_34 + c_1_02·c_1_23·c_1_33 + c_1_02·c_1_25·c_1_3
       + c_1_02·c_1_26 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_24 + 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