Cohomology of group number 123 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 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 3.
  • Its center has rank 1.
  • 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 exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    ( − 1) · (t5  −  t4  +  2·t2  −  t  +  1)
    (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-∞,-3,-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 19 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. b_2_1, an element of degree 2
  5. b_2_2, an element of degree 2
  6. b_2_3, an element of degree 2
  7. a_3_3, a nilpotent element of degree 3
  8. a_3_4, a nilpotent element of degree 3
  9. b_3_5, an element of degree 3
  10. b_3_6, an element of degree 3
  11. b_3_7, an element of degree 3
  12. b_4_9, an element of degree 4
  13. b_4_10, an element of degree 4
  14. b_5_11, an element of degree 5
  15. b_5_13, an element of degree 5
  16. a_6_8, a nilpotent element of degree 6
  17. b_6_17, an element of degree 6
  18. b_7_18, an element of degree 7
  19. c_8_26, 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 116 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_1
  5. a_2_0·a_1_0
  6. b_2_2·a_1_1 + b_2_1·a_1_1
  7. b_2_2·a_1_0 + b_2_1·a_1_1
  8. b_2_3·a_1_0
  9. a_2_02
  10. b_2_22 + b_2_1·b_2_3 + b_2_1·b_2_2
  11. a_2_0·b_2_2 + a_1_1·a_3_3
  12. a_2_0·b_2_1 + a_1_0·a_3_3
  13. a_1_1·a_3_4
  14. a_2_0·b_2_2 + a_2_0·b_2_1 + a_1_0·a_3_4
  15. a_1_1·b_3_5 + a_2_0·b_2_3
  16. a_1_0·b_3_5 + a_2_0·b_2_2 + a_2_0·b_2_1
  17. b_2_2·b_2_3 + a_1_1·b_3_6 + a_2_0·b_2_3
  18. b_2_22 + b_2_1·b_2_2 + a_1_0·b_3_6
  19. b_2_2·b_2_3 + a_1_0·b_3_7 + a_2_0·b_2_1
  20. a_2_0·a_3_3
  21. b_2_2·a_3_3 + b_2_1·a_3_4 + b_2_1·a_3_3 + b_2_12·a_1_1
  22. b_2_3·a_3_3 + b_2_2·a_3_4 + b_2_12·a_1_1
  23. a_2_0·a_3_4
  24. b_2_1·b_3_5 + b_2_2·a_3_3 + b_2_1·a_3_3
  25. b_2_2·b_3_5 + b_2_3·a_3_3
  26. b_2_32·a_1_1 + a_2_0·b_3_5
  27. b_2_3·b_3_6 + b_2_3·b_3_5 + b_2_3·a_3_4
  28. b_2_3·a_3_3 + b_2_32·a_1_1 + a_2_0·b_3_6
  29. b_2_2·b_3_6 + b_2_1·b_3_7 + b_2_1·a_3_3
  30. b_2_2·b_3_7 + b_2_2·b_3_6 + b_2_2·a_3_3
  31. b_2_3·a_3_4 + b_2_3·a_3_3 + a_2_0·b_3_7
  32. b_4_9·a_1_1 + b_2_3·a_3_4 + b_2_3·a_3_3 + b_2_32·a_1_1 + b_2_12·a_1_1
  33. b_4_9·a_1_0 + b_2_3·a_3_3 + b_2_12·a_1_1
  34. b_4_10·a_1_0 + b_2_12·a_1_1
  35. a_3_42 + a_3_3·a_3_4 + b_2_1·a_1_0·a_3_4 + b_2_1·a_1_0·a_3_3
  36. a_3_3·b_3_5 + a_3_42
  37. b_3_52 + b_2_33 + a_2_0·b_2_32 + a_3_42
  38. b_2_1·a_1_0·b_3_6 + a_3_32
  39. b_3_5·b_3_6 + b_2_33 + a_3_4·b_3_6 + a_2_0·b_2_32 + a_3_42 + a_3_32
  40. b_2_1·a_1_0·b_3_7 + a_3_42 + a_3_32 + b_2_1·a_1_0·a_3_3
  41. a_3_4·b_3_5 + b_2_3·a_1_1·b_3_7 + a_3_42
  42. a_3_4·b_3_6 + a_3_4·b_3_5 + a_3_3·b_3_7 + a_3_3·b_3_6
  43. b_2_1·b_4_9 + b_2_12·b_2_2 + a_3_3·b_3_6 + a_3_42
  44. b_3_5·b_3_7 + b_2_3·b_4_9 + b_2_33 + a_3_4·b_3_5 + a_3_42 + a_3_32
  45. b_2_2·b_4_9 + b_2_12·b_2_2 + a_3_4·b_3_6 + a_3_4·b_3_5 + a_3_3·b_3_6
  46. a_3_4·b_3_5 + a_2_0·b_4_9 + a_2_0·b_2_32 + a_3_42 + b_2_1·a_1_0·a_3_4
       + b_2_1·a_1_0·a_3_3
  47. b_3_62 + b_2_33 + b_2_1·b_4_10 + b_2_12·b_2_2 + a_2_0·b_2_32 + a_3_32
       + b_2_1·a_1_0·a_3_4 + b_2_1·a_1_0·a_3_3
  48. b_3_72 + b_3_6·b_3_7 + b_3_5·b_3_7 + b_2_3·b_4_10 + b_2_33 + a_3_3·b_3_6
       + a_2_0·b_2_32
  49. b_3_6·b_3_7 + b_3_5·b_3_7 + b_2_2·b_4_10 + b_2_12·b_2_2 + a_3_4·b_3_7 + a_3_3·b_3_6
       + a_3_32
  50. a_3_4·b_3_7 + a_2_0·b_4_10 + a_2_0·b_2_32 + a_3_32
  51. a_3_4·b_3_7 + a_1_1·b_5_11 + a_2_0·b_2_32 + a_3_42
  52. a_1_0·b_5_11 + a_3_32 + b_2_1·a_1_0·a_3_4 + b_2_1·a_1_0·a_3_3
  53. a_3_4·b_3_5 + a_1_1·b_5_13 + a_2_0·b_2_32 + a_3_42 + b_2_1·a_1_0·a_3_4
       + b_2_1·a_1_0·a_3_3
  54. a_1_0·b_5_13 + b_2_1·a_1_0·a_3_4 + b_2_1·a_1_0·a_3_3
  55. b_4_9·a_3_3 + b_2_12·a_3_4 + b_2_12·a_3_3 + b_2_13·a_1_1 + a_1_0·a_3_3·b_3_7
       + a_1_0·a_3_3·b_3_6
  56. b_4_9·b_3_5 + b_2_32·b_3_7 + b_2_32·b_3_5 + a_1_0·a_3_3·b_3_7
  57. b_4_9·a_3_4 + b_2_3·b_4_10·a_1_1 + b_2_13·a_1_1 + a_2_0·b_2_3·b_3_7 + a_2_0·b_2_3·b_3_5
  58. b_4_9·b_3_6 + b_2_32·b_3_7 + b_2_32·b_3_5 + b_2_12·b_3_7 + b_4_10·a_3_3 + b_4_9·a_3_4
       + b_2_12·a_3_4 + a_1_0·a_3_3·b_3_6
  59. b_4_10·b_3_5 + b_4_9·b_3_7 + b_4_9·b_3_6 + b_4_9·a_3_4 + b_2_12·a_3_4 + b_2_12·a_3_3
       + a_2_0·b_2_3·b_3_5 + a_1_0·a_3_3·b_3_6
  60. b_2_2·b_5_11 + b_2_12·b_3_7 + b_2_12·a_3_4 + b_2_13·a_1_1 + a_1_0·a_3_3·b_3_6
  61. b_4_9·a_3_4 + b_2_13·a_1_1 + a_2_0·b_5_11 + a_2_0·b_2_3·b_3_7 + a_2_0·b_2_3·b_3_5
       + a_1_0·a_3_3·b_3_6
  62. b_2_3·b_5_13 + b_2_32·b_3_7 + b_2_32·b_3_5 + a_2_0·b_2_3·b_3_7 + a_2_0·b_2_3·b_3_5
       + a_1_0·a_3_3·b_3_7
  63. b_2_2·b_5_13 + b_2_1·b_5_13 + b_2_1·b_5_11 + b_2_12·b_3_6 + b_2_12·a_3_4 + b_2_12·a_3_3
       + b_2_13·a_1_1 + a_1_0·a_3_3·b_3_6
  64. a_2_0·b_5_13 + a_2_0·b_2_3·b_3_7 + a_2_0·b_2_3·b_3_5
  65. a_1_0·a_3_3·b_3_7 + a_6_8·a_1_1
  66. a_1_0·a_3_3·b_3_7 + a_6_8·a_1_0
  67. b_4_9·b_3_7 + b_4_9·b_3_6 + b_2_3·b_5_11 + b_6_17·a_1_1 + b_4_10·a_3_4 + b_2_12·a_3_4
       + b_2_12·a_3_3 + b_2_13·a_1_1 + a_2_0·b_2_3·b_3_7 + a_1_0·a_3_3·b_3_6
  68. b_6_17·a_1_0 + b_2_13·a_1_1 + a_1_0·a_3_3·b_3_7
  69. b_4_92 + b_2_32·b_4_10 + b_2_13·b_2_2 + a_2_0·b_2_33 + b_2_1·a_3_32
  70. a_3_4·b_5_11 + a_3_3·b_5_11 + b_4_10·a_1_1·b_3_7 + b_2_1·a_3_3·b_3_7 + b_2_1·a_3_3·a_3_4
       + b_2_12·a_1_0·a_3_4 + b_2_12·a_1_0·a_3_3
  71. a_3_4·b_5_13 + a_3_4·b_5_11 + b_4_10·a_1_1·b_3_7 + b_2_1·a_3_3·b_3_7 + b_2_1·a_3_3·b_3_6
       + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_3·b_4_9 + b_2_1·a_3_32 + b_2_12·a_1_0·a_3_4
       + b_2_12·a_1_0·a_3_3
  72. b_3_5·b_5_13 + b_2_32·b_4_9 + a_3_4·b_5_11 + b_4_10·a_1_1·b_3_7 + b_2_1·a_3_3·b_3_7
       + b_2_1·a_3_3·b_3_6 + b_2_1·a_3_32
  73. b_3_7·b_5_13 + b_3_6·b_5_13 + b_3_6·b_5_11 + b_3_5·b_5_11 + b_4_92 + b_2_12·b_4_10
       + a_3_4·b_5_11 + a_3_3·b_5_13 + b_2_1·a_3_3·b_3_7 + a_2_0·b_2_3·b_4_10
       + a_2_0·b_2_3·b_4_9 + b_2_1·a_3_3·a_3_4 + b_2_12·a_1_0·a_3_4 + b_2_12·a_1_0·a_3_3
  74. a_3_3·b_5_13 + b_2_1·a_3_3·b_3_7 + b_2_1·a_6_8 + b_2_12·a_1_0·a_3_4
  75. b_3_5·b_5_11 + b_4_92 + b_2_13·b_2_2 + a_3_4·b_5_11 + b_4_10·a_1_1·b_3_7 + b_2_3·a_6_8
       + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_3·b_4_9 + b_2_1·a_3_32
  76. a_3_4·b_5_11 + a_3_3·b_5_13 + b_4_10·a_1_1·b_3_7 + b_2_2·a_6_8 + b_2_1·a_3_3·b_3_6
       + b_2_1·a_3_3·a_3_4 + b_2_1·a_3_32 + b_2_12·a_1_0·a_3_4 + b_2_12·a_1_0·a_3_3
  77. a_2_0·a_6_8
  78. b_3_6·b_5_13 + b_2_32·b_4_9 + b_2_1·b_6_17 + b_2_13·b_2_2 + a_2_0·b_2_3·b_4_10
       + a_2_0·b_2_3·b_4_9 + b_2_1·a_3_3·a_3_4 + b_2_1·a_3_32 + b_2_12·a_1_0·a_3_4
       + b_2_12·a_1_0·a_3_3
  79. b_3_6·b_5_13 + b_3_6·b_5_11 + b_3_5·b_5_11 + b_2_32·b_4_9 + b_2_2·b_6_17 + b_2_12·b_4_10
       + a_3_4·b_5_11 + b_2_1·a_3_3·b_3_7 + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_3·b_4_9
       + b_2_1·a_3_32
  80. b_3_5·b_5_11 + b_4_92 + b_2_13·b_2_2 + a_3_4·b_5_11 + a_2_0·b_6_17 + a_2_0·b_2_33
       + b_2_1·a_3_3·a_3_4 + b_2_12·a_1_0·a_3_4 + b_2_12·a_1_0·a_3_3
  81. b_3_5·b_5_11 + b_4_92 + b_2_13·b_2_2 + a_3_4·b_5_11 + a_1_1·b_7_18 + b_2_1·a_3_3·a_3_4
       + b_2_12·a_1_0·a_3_4 + b_2_12·a_1_0·a_3_3
  82. a_1_0·b_7_18 + b_2_12·a_1_0·a_3_3
  83. a_6_8·a_3_4 + a_3_33
  84. a_6_8·a_3_3 + a_3_32·a_3_4
  85. b_4_9·b_5_13 + b_2_32·b_5_11 + b_2_12·b_5_13 + b_2_12·b_5_11 + b_2_13·b_3_6
       + a_6_8·b_3_6 + b_2_1·b_4_10·a_3_4 + b_2_1·b_4_10·a_3_3 + b_2_14·a_1_1
       + a_2_0·b_2_3·b_5_11 + a_2_0·b_2_32·b_3_7 + a_2_0·b_2_32·b_3_5 + a_3_33
  86. b_4_9·b_5_13 + b_4_9·b_5_11 + b_2_3·b_4_10·b_3_7 + b_2_12·b_5_13 + b_2_12·b_5_11
       + b_2_13·b_3_7 + b_2_13·b_3_6 + a_6_8·b_3_7 + a_6_8·b_3_5 + b_2_1·b_4_10·a_3_4
       + b_2_13·a_3_3 + b_2_14·a_1_1 + a_2_0·b_2_32·b_3_7
  87. a_6_8·b_3_5 + b_2_3·b_6_17·a_1_1 + a_2_0·b_4_10·b_3_7 + a_2_0·b_2_3·b_5_11
       + a_2_0·b_2_32·b_3_7 + a_3_33
  88. b_4_9·b_5_11 + b_2_3·b_4_10·b_3_7 + b_2_32·b_5_11 + b_2_13·b_3_7 + b_6_17·a_3_4
       + b_4_102·a_1_1 + b_2_1·b_4_10·a_3_3 + b_2_13·a_3_3 + a_2_0·b_4_10·b_3_7
       + a_2_0·b_2_3·b_5_11 + a_3_32·a_3_4
  89. b_4_9·b_5_13 + b_2_32·b_5_11 + b_2_12·b_5_13 + b_2_12·b_5_11 + b_2_13·b_3_6
       + b_6_17·a_3_3 + a_6_8·b_3_5 + a_2_0·b_2_3·b_5_11 + a_2_0·b_2_32·b_3_7
       + a_2_0·b_2_32·b_3_5
  90. b_6_17·b_3_6 + b_6_17·b_3_5 + b_4_10·b_5_13 + b_4_9·b_5_11 + b_2_12·b_5_13
       + b_2_12·b_5_11 + b_2_13·b_3_6 + a_6_8·b_3_5 + b_4_102·a_1_1 + b_2_1·b_4_10·a_3_3
       + b_2_13·a_3_4 + b_2_13·a_3_3 + a_2_0·b_2_32·b_3_7 + a_3_33
  91. b_2_1·b_7_18 + b_2_1·b_4_10·b_3_7 + b_2_1·b_4_10·b_3_6 + b_2_12·b_5_11 + b_2_13·b_3_6
       + b_2_1·b_4_10·a_3_3 + b_2_13·a_3_4 + b_2_14·a_1_1 + a_3_32·a_3_4 + a_3_33
  92. b_6_17·b_3_5 + b_4_9·b_5_11 + b_2_3·b_7_18 + b_2_3·b_4_10·b_3_7 + b_2_32·b_5_11
       + b_2_33·b_3_5 + b_2_13·b_3_7 + a_6_8·b_3_5 + b_2_1·b_4_10·a_3_3 + b_2_13·a_3_3
       + a_2_0·b_2_32·b_3_7 + a_3_33
  93. b_2_2·b_7_18 + b_2_13·a_3_4 + b_2_13·a_3_3 + b_2_14·a_1_1 + a_3_32·a_3_4 + a_3_33
  94. a_6_8·b_3_5 + a_2_0·b_7_18 + a_2_0·b_4_10·b_3_7 + a_2_0·b_2_3·b_5_11
       + a_2_0·b_2_32·b_3_7 + a_2_0·b_2_32·b_3_5 + a_3_33
  95. b_5_112 + b_2_3·b_4_102 + b_2_2·b_4_102 + b_2_1·b_4_102 + b_2_13·b_4_10
       + b_2_14·b_2_2 + a_2_0·b_4_102 + b_2_12·a_3_3·a_3_4 + b_2_13·a_1_0·a_3_4
       + b_2_13·a_1_0·a_3_3
  96. b_5_132 + b_2_33·b_4_10 + b_2_1·b_4_102 + b_2_14·b_2_2 + a_2_0·b_2_32·b_4_10
       + a_2_0·b_2_34 + b_2_12·a_3_3·a_3_4 + b_2_13·a_1_0·a_3_4 + b_2_13·a_1_0·a_3_3
  97. b_5_11·b_5_13 + b_2_3·b_4_9·b_4_10 + b_2_2·b_4_102 + b_2_1·b_4_102 + b_2_12·b_6_17
       + b_2_14·b_2_2 + b_4_9·a_6_8 + b_2_12·a_3_3·b_3_7 + a_2_0·b_2_32·b_4_10
       + a_2_0·b_2_32·b_4_9 + b_2_12·a_3_32
  98. b_6_17·a_1_1·b_3_7 + b_4_9·a_6_8 + b_2_1·b_2_2·a_6_8 + a_2_0·b_4_102
       + a_2_0·b_2_3·b_6_17 + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_32·b_4_9 + b_2_12·a_3_3·a_3_4
       + b_2_12·a_3_32 + b_2_13·a_1_0·a_3_4 + b_2_13·a_1_0·a_3_3
  99. a_3_4·b_7_18 + b_4_10·a_3_3·b_3_7 + b_4_10·a_3_3·b_3_6 + b_4_9·a_6_8 + b_2_12·a_6_8
       + a_2_0·b_4_102 + a_2_0·b_2_3·b_6_17 + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_34
       + b_2_12·a_3_3·a_3_4 + b_2_12·a_3_32 + b_2_13·a_1_0·a_3_3
  100. a_3_3·b_7_18 + b_4_10·a_3_3·b_3_7 + b_4_10·a_3_3·b_3_6 + b_2_1·b_2_2·a_6_8
       + b_2_12·a_6_8 + b_2_13·a_1_0·a_3_4
  101. b_3_5·b_7_18 + b_2_32·b_6_17 + b_2_35 + b_4_10·a_3_3·b_3_7 + b_4_10·a_3_3·b_3_6
       + b_4_9·a_6_8 + b_2_12·a_6_8 + a_2_0·b_2_3·b_6_17 + a_2_0·b_2_32·b_4_10
       + a_2_0·b_2_32·b_4_9 + a_2_0·b_2_34 + b_2_12·a_3_3·a_3_4 + b_2_12·a_3_32
       + b_2_13·a_1_0·a_3_3
  102. b_5_11·b_5_13 + b_3_6·b_7_18 + b_2_3·b_4_9·b_4_10 + b_2_32·b_6_17 + b_2_35
       + b_2_1·b_2_2·b_6_17 + b_2_14·b_2_2 + b_4_9·a_6_8 + b_2_12·a_3_3·b_3_7
       + b_2_12·a_3_3·b_3_6 + a_2_0·b_4_102 + a_2_0·b_2_32·b_4_10 + b_2_12·a_3_3·a_3_4
       + b_2_12·a_3_32
  103. b_3_7·b_7_18 + b_4_9·b_6_17 + b_2_32·b_6_17 + b_2_33·b_4_9 + b_2_35
       + b_2_1·b_2_2·b_6_17 + b_4_10·a_3_3·b_3_6 + b_4_10·a_6_8 + b_2_12·a_3_3·b_3_7
       + b_2_12·a_6_8 + a_2_0·b_4_102 + a_2_0·b_4_9·b_4_10 + a_2_0·b_2_3·b_6_17
       + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_32·b_4_9 + a_2_0·b_2_34 + b_2_12·a_3_3·a_3_4
       + b_2_13·a_1_0·a_3_3
  104. a_6_8·b_5_11 + b_4_10·b_6_17·a_1_1 + b_4_102·a_3_4 + b_2_1·a_6_8·b_3_6
       + a_2_0·b_4_10·b_5_11 + a_2_0·b_2_3·b_4_10·b_3_7 + b_2_1·a_3_32·a_3_4 + b_2_1·a_3_33
  105. b_6_17·b_5_13 + b_4_102·b_3_6 + b_2_3·b_6_17·b_3_7 + b_2_3·b_4_10·b_5_11
       + b_2_32·b_7_18 + b_2_34·b_3_5 + b_2_13·b_5_13 + b_2_13·b_5_11 + b_2_14·b_3_7
       + b_2_14·b_3_6 + a_6_8·b_5_11 + b_4_102·a_3_4 + b_2_1·a_6_8·b_3_6 + b_2_14·a_3_4
       + a_2_0·b_4_10·b_5_11 + a_2_0·b_2_3·b_4_10·b_3_7 + b_2_1·a_3_32·a_3_4
  106. a_6_8·b_5_13 + b_4_102·a_3_3 + b_2_1·a_6_8·b_3_7 + b_2_12·b_4_10·a_3_4
       + b_2_12·b_4_10·a_3_3 + b_2_15·a_1_1 + a_2_0·b_6_17·b_3_7 + a_2_0·b_4_10·b_5_11
       + a_2_0·b_2_3·b_7_18 + a_2_0·b_2_32·b_5_11 + a_2_0·b_2_33·b_3_7 + b_2_1·a_3_33
  107. b_6_17·b_5_13 + b_4_102·b_3_6 + b_4_9·b_7_18 + b_2_3·b_4_10·b_5_11 + b_2_34·b_3_7
       + b_2_34·b_3_5 + b_2_13·b_5_13 + b_2_13·b_5_11 + b_2_14·b_3_7 + b_2_14·b_3_6
       + a_6_8·b_5_13 + b_4_102·a_3_4 + b_4_102·a_3_3 + b_2_1·a_6_8·b_3_6
       + b_2_12·b_4_10·a_3_4 + b_2_12·b_4_10·a_3_3 + b_2_14·a_3_3 + a_2_0·b_4_10·b_5_11
       + a_2_0·b_2_32·b_5_11 + a_2_0·b_2_33·b_3_7 + a_2_0·b_2_33·b_3_5 + b_2_1·a_3_33
  108. b_6_17·b_5_11 + b_4_10·b_7_18 + b_2_33·b_5_11 + b_2_1·b_4_10·b_5_13
       + b_2_1·b_4_10·b_5_11 + b_2_12·b_4_10·b_3_6 + b_2_13·b_5_13 + b_2_13·b_5_11
       + b_2_14·b_3_7 + b_2_14·b_3_6 + a_6_8·b_5_13 + a_6_8·b_5_11 + b_4_102·a_3_4
       + b_4_102·a_3_3 + b_2_1·a_6_8·b_3_6 + b_2_12·b_4_10·a_3_4 + b_2_14·a_3_4
       + b_2_15·a_1_1 + a_2_0·b_6_17·b_3_7 + a_2_0·b_4_10·b_5_11 + a_2_0·b_2_3·b_4_10·b_3_7
       + a_2_0·b_2_33·b_3_7 + b_2_1·a_3_33 + b_2_3·c_8_26·a_1_1
  109. a_6_82
  110. b_5_11·b_7_18 + b_4_10·b_3_7·b_5_11 + b_4_9·b_4_102 + b_2_3·b_4_10·b_6_17
       + b_2_32·b_4_102 + b_2_34·b_4_10 + b_2_2·b_4_10·b_6_17 + b_2_1·b_4_10·b_6_17
       + b_2_12·b_2_2·b_6_17 + b_2_13·b_6_17 + b_2_13·b_2_2·b_4_10 + a_6_8·b_6_17
       + b_2_1·b_4_10·a_3_3·b_3_6 + b_2_1·b_4_10·a_6_8 + b_2_12·b_2_2·a_6_8
       + b_2_13·a_3_3·b_3_6 + b_2_13·a_6_8 + a_2_0·b_4_9·b_6_17 + a_2_0·b_2_3·b_4_102
       + a_2_0·b_2_3·b_4_9·b_4_10 + a_2_0·b_2_33·b_4_10 + b_2_13·a_3_3·a_3_4
       + b_2_13·a_3_32 + b_2_14·a_1_0·a_3_3
  111. b_5_13·b_7_18 + b_2_3·b_4_9·b_6_17 + b_2_34·b_4_9 + b_2_2·b_4_10·b_6_17
       + b_2_1·b_4_10·b_6_17 + b_2_1·b_2_2·b_4_102 + b_2_12·b_4_102 + b_2_3·b_4_10·a_6_8
       + b_2_13·a_3_3·b_3_7 + b_2_13·a_6_8 + a_2_0·b_2_3·b_4_9·b_4_10 + a_2_0·b_2_33·b_4_10
       + b_2_13·a_3_3·a_3_4 + b_2_13·a_3_32 + b_2_14·a_1_0·a_3_3
  112. b_6_172 + b_4_103 + b_2_3·b_4_10·b_6_17 + b_2_32·b_4_102 + b_2_32·b_4_9·b_4_10
       + b_2_34·b_4_10 + b_2_36 + b_2_1·b_2_2·b_4_102 + b_2_13·b_2_2·b_4_10
       + b_2_3·b_4_10·a_6_8 + a_2_0·b_4_10·b_6_17 + a_2_0·b_4_9·b_6_17 + a_2_0·b_2_3·b_4_102
       + a_2_0·b_2_3·b_4_9·b_4_10 + a_2_0·b_2_32·b_6_17 + a_2_0·b_2_33·b_4_9
       + b_2_13·a_3_3·a_3_4 + b_2_13·a_3_32 + b_2_14·a_1_0·a_3_4 + b_2_14·a_1_0·a_3_3
       + b_2_32·c_8_26
  113. b_4_10·b_3_7·b_5_11 + b_4_9·b_4_102 + b_2_32·b_4_102 + b_2_13·b_2_2·b_4_10
       + a_6_8·b_6_17 + b_2_1·b_4_10·a_3_3·b_3_6 + b_2_1·b_4_10·a_6_8 + a_2_0·b_4_9·b_6_17
       + a_2_0·b_2_3·b_4_9·b_4_10 + a_2_0·b_2_32·b_6_17 + a_2_0·b_2_33·b_4_10
       + a_2_0·b_2_35 + b_2_13·a_3_3·a_3_4 + b_2_13·a_3_32 + b_2_14·a_1_0·a_3_4
       + b_2_14·a_1_0·a_3_3 + a_2_0·b_2_3·c_8_26
  114. a_6_8·b_7_18 + b_4_10·a_6_8·b_3_7 + b_4_10·a_6_8·b_3_6 + b_2_1·b_4_102·a_3_4
       + b_2_16·a_1_1 + a_2_0·b_4_10·b_7_18 + a_2_0·b_2_3·b_6_17·b_3_7 + a_2_0·b_2_32·b_7_18
       + a_2_0·b_2_32·b_4_10·b_3_7 + a_2_0·b_2_33·b_5_11 + a_2_0·b_2_34·b_3_7
       + b_2_12·a_3_33 + a_2_0·c_8_26·b_3_5
  115. b_6_17·b_7_18 + b_4_102·b_5_11 + b_2_3·b_4_10·b_7_18 + b_2_32·b_4_10·b_5_11
       + b_2_33·b_7_18 + b_2_33·b_4_10·b_3_7 + b_2_34·b_5_11 + b_2_1·b_4_102·b_3_7
       + b_4_103·a_1_1 + b_2_1·b_4_102·a_3_4 + b_2_12·a_6_8·b_3_6 + b_2_13·b_4_10·a_3_4
       + b_2_13·b_4_10·a_3_3 + b_2_16·a_1_1 + a_2_0·b_4_10·b_7_18 + a_2_0·b_4_102·b_3_7
       + b_2_3·c_8_26·b_3_5 + a_2_0·c_8_26·b_3_7 + a_2_0·c_8_26·b_3_6
  116. b_7_182 + b_2_3·b_4_103 + b_2_32·b_4_10·b_6_17 + b_2_33·b_4_102
       + b_2_33·b_4_9·b_4_10 + b_2_35·b_4_10 + b_2_2·b_4_103 + b_2_1·b_4_103
       + b_2_12·b_2_2·b_4_102 + b_2_13·b_4_102 + a_2_0·b_4_103 + a_2_0·b_4_9·b_4_102
       + a_2_0·b_2_3·b_4_10·b_6_17 + a_2_0·b_2_3·b_4_9·b_6_17 + a_2_0·b_2_32·b_4_9·b_4_10
       + a_2_0·b_2_33·b_6_17 + a_2_0·b_2_34·b_4_9 + b_2_14·a_3_3·a_3_4
       + b_2_15·a_1_0·a_3_4 + b_2_15·a_1_0·a_3_3 + b_2_33·c_8_26 + a_2_0·b_2_32·c_8_26

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_8_26, a Duflot regular element of degree 8
    2. b_4_10 + b_2_1·b_2_2 + b_2_12, an element of degree 4
    3. b_2_3·b_4_10 + b_2_33 + b_2_1·b_4_10 + b_2_12·b_2_2, an element of degree 6
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 15].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
  • We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 2 elements of degree 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 1

  1. a_1_0 → 0, an element of degree 1
  2. a_1_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_5 → 0, an element of degree 3
  10. b_3_6 → 0, an element of degree 3
  11. b_3_7 → 0, an element of degree 3
  12. b_4_9 → 0, an element of degree 4
  13. b_4_10 → 0, an element of degree 4
  14. b_5_11 → 0, an element of degree 5
  15. b_5_13 → 0, an element of degree 5
  16. a_6_8 → 0, an element of degree 6
  17. b_6_17 → 0, an element of degree 6
  18. b_7_18 → 0, an element of degree 7
  19. c_8_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_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. b_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → c_1_12, an element of degree 2
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_5 → c_1_13, an element of degree 3
  10. b_3_6 → c_1_13, an element of degree 3
  11. b_3_7 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  12. b_4_9 → c_1_12·c_1_22 + c_1_13·c_1_2 + c_1_14, an element of degree 4
  13. b_4_10 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. b_5_11 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_15, an element of degree 5
  15. b_5_13 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_15, an element of degree 5
  16. a_6_8 → 0, an element of degree 6
  17. b_6_17 → c_1_26 + c_1_15·c_1_2 + c_1_16 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  18. b_7_18 → c_1_1·c_1_26 + c_1_16·c_1_2 + c_1_02·c_1_15 + c_1_04·c_1_13, an element of degree 7
  19. c_8_26 → c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + 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_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. b_2_1 → c_1_12, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_5 → 0, an element of degree 3
  10. b_3_6 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  11. b_3_7 → 0, an element of degree 3
  12. b_4_9 → 0, an element of degree 4
  13. b_4_10 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  14. b_5_11 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
  15. b_5_13 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  16. a_6_8 → 0, an element of degree 6
  17. b_6_17 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23, an element of degree 6
  18. b_7_18 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_15·c_1_22, an element of degree 7
  19. c_8_26 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·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
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + 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_1 → 0, an element of degree 1
  3. a_2_0 → 0, an element of degree 2
  4. b_2_1 → c_1_22, an element of degree 2
  5. b_2_2 → c_1_22, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. b_3_5 → 0, an element of degree 3
  10. b_3_6 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  11. b_3_7 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  12. b_4_9 → c_1_24, an element of degree 4
  13. b_4_10 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  14. b_5_11 → c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
  15. b_5_13 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
  16. a_6_8 → 0, an element of degree 6
  17. b_6_17 → c_1_26 + c_1_13·c_1_23 + c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_16, an element of degree 6
  18. b_7_18 → 0, an element of degree 7
  19. c_8_26 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + 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