Cohomology of group number 25 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 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

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
    t5  −  t4  +  2·t2  −  t  +  1
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • 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 20 minimal generators of maximal degree 6:

  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_2, a nilpotent element of degree 3
  8. a_3_3, a nilpotent element of degree 3
  9. b_3_4, an element of degree 3
  10. b_3_5, an element of degree 3
  11. b_3_6, an element of degree 3
  12. b_3_7, an element of degree 3
  13. a_4_3, a nilpotent element of degree 4
  14. a_4_4, a nilpotent element of degree 4
  15. b_4_10, an element of degree 4
  16. c_4_11, a Duflot regular element of degree 4
  17. c_4_12, a Duflot regular element of degree 4
  18. b_5_16, an element of degree 5
  19. b_5_17, an element of degree 5
  20. a_6_7, a nilpotent element of degree 6

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

Ring relations

There are 132 minimal relations of maximal degree 12:

  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_1·a_1_0
  7. b_2_2·a_1_1 + b_2_1·a_1_1
  8. b_2_2·a_1_0 + b_2_1·a_1_1
  9. b_2_3·a_1_0
  10. a_2_02
  11. b_2_22 + b_2_1·b_2_3 + b_2_1·b_2_2
  12. a_2_0·b_2_2 + a_1_1·a_3_2
  13. a_2_0·b_2_1 + a_1_0·a_3_2
  14. a_2_0·b_2_2 + a_2_0·b_2_1 + a_1_1·a_3_3
  15. a_2_0·b_2_2 + a_1_0·a_3_3
  16. a_1_1·b_3_4 + a_2_0·b_2_2
  17. a_1_0·b_3_4 + a_2_0·b_2_1
  18. a_1_1·b_3_5
  19. a_1_0·b_3_5
  20. a_1_1·b_3_6
  21. a_1_0·b_3_6 + a_2_0·b_2_2 + a_2_0·b_2_1
  22. a_1_1·b_3_7 + a_2_0·b_2_3 + a_2_0·b_2_2
  23. a_1_0·b_3_7 + a_2_0·b_2_2
  24. a_2_0·a_3_2
  25. b_2_2·a_3_2 + b_2_1·a_3_3
  26. b_2_3·a_3_2 + b_2_2·a_3_3 + b_2_2·a_3_2
  27. a_2_0·a_3_3
  28. a_2_0·b_3_4
  29. b_2_3·b_3_5 + b_2_3·b_3_4 + b_2_1·b_3_5 + b_2_3·a_3_2
  30. a_2_0·b_3_5
  31. b_2_2·b_3_5 + b_2_2·b_3_4 + b_2_1·b_3_6 + b_2_1·b_3_4
  32. b_2_2·b_3_6 + b_2_2·b_3_5 + b_2_1·b_3_5 + b_2_3·a_3_2
  33. a_2_0·b_3_6
  34. b_2_2·b_3_4 + b_2_1·b_3_7
  35. b_2_3·b_3_4 + b_2_2·b_3_7 + b_2_2·b_3_4
  36. b_2_32·a_1_1 + a_2_0·b_3_7
  37. a_4_3·a_1_1
  38. a_4_3·a_1_0
  39. a_4_4·a_1_1
  40. a_4_4·a_1_0
  41. b_2_3·b_3_6 + b_2_3·b_3_4 + b_2_2·b_3_5 + b_4_10·a_1_1 + b_2_3·a_3_3 + b_2_32·a_1_1
  42. b_4_10·a_1_0
  43. a_3_22
  44. a_3_2·a_3_3
  45. a_3_32
  46. b_3_42 + b_2_1·b_2_32 + b_2_13
  47. b_3_4·b_3_5 + b_2_1·b_2_32 + b_2_12·b_2_3 + a_3_2·b_3_5
  48. b_3_52 + b_2_1·b_2_32
  49. a_3_3·b_3_5 + a_3_3·b_3_4 + a_3_2·b_3_6 + a_3_2·b_3_4
  50. a_3_3·b_3_6 + a_3_3·b_3_5 + a_3_2·b_3_5
  51. b_3_4·b_3_6 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_2 + b_2_13 + a_3_3·b_3_5
  52. b_3_5·b_3_6 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3 + a_3_3·b_3_5
       + a_3_2·b_3_5
  53. b_3_62 + b_2_1·b_2_32 + b_2_12·b_2_3 + b_2_12·b_2_2 + b_2_13
  54. a_3_3·b_3_4 + a_3_2·b_3_7
  55. b_3_4·b_3_7 + b_2_2·b_2_32 + b_2_12·b_2_2
  56. b_3_5·b_3_7 + b_2_2·b_2_32 + b_2_1·b_2_2·b_2_3 + a_3_3·b_3_5
  57. b_3_72 + b_2_33 + b_2_2·b_2_32 + b_2_12·b_2_3 + b_2_12·b_2_2 + a_2_0·b_2_32
  58. a_3_2·b_3_5 + a_3_2·b_3_4 + b_2_1·a_4_3
  59. b_3_6·b_3_7 + b_2_2·b_2_32 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3
       + a_3_3·b_3_7 + a_3_3·b_3_5 + a_3_3·b_3_4 + b_2_3·a_4_3
  60. a_3_3·b_3_5 + a_3_3·b_3_4 + b_2_2·a_4_3
  61. a_2_0·a_4_3
  62. a_3_2·b_3_4 + b_2_1·a_4_4
  63. a_3_3·b_3_7 + a_3_3·b_3_4 + b_2_3·a_4_4 + a_2_0·b_2_32
  64. a_3_3·b_3_4 + b_2_2·a_4_4
  65. a_2_0·a_4_4
  66. b_3_6·b_3_7 + b_2_2·b_2_32 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3
       + a_3_3·b_3_7 + a_3_3·b_3_5 + a_3_3·b_3_4 + a_3_2·b_3_5 + a_2_0·b_4_10 + a_2_0·b_2_32
  67. a_1_1·b_5_16 + a_2_0·b_2_32
  68. a_1_0·b_5_16
  69. b_3_6·b_3_7 + b_2_2·b_2_32 + b_2_1·b_2_32 + b_2_1·b_2_2·b_2_3 + b_2_12·b_2_3
       + a_3_3·b_3_7 + a_3_3·b_3_5 + a_3_3·b_3_4 + a_3_2·b_3_5 + a_1_1·b_5_17 + a_2_0·b_2_32
  70. a_1_0·b_5_17
  71. a_4_3·a_3_2
  72. a_4_3·a_3_3
  73. a_4_3·b_3_4 + b_2_1·b_2_2·a_3_3 + b_2_12·a_3_3 + b_2_12·a_3_2
  74. a_4_3·b_3_5 + b_2_1·b_2_2·a_3_3 + b_2_12·a_3_3
  75. a_4_3·b_3_6 + b_2_1·b_2_2·a_3_3 + b_2_12·a_3_2
  76. a_4_4·a_3_2
  77. a_4_4·a_3_3
  78. a_4_4·b_3_4 + b_2_2·b_2_3·a_3_3 + b_2_1·b_2_3·a_3_3 + b_2_12·a_3_2
  79. a_4_4·b_3_5 + b_2_2·b_2_3·a_3_3 + b_2_1·b_2_3·a_3_3 + b_2_1·b_2_2·a_3_3 + b_2_12·a_3_3
  80. a_4_4·b_3_6 + b_2_2·b_2_3·a_3_3 + b_2_12·a_3_3 + b_2_12·a_3_2
  81. a_4_4·b_3_7 + b_2_32·a_3_3 + b_2_12·a_3_3 + a_2_0·b_2_3·b_3_7
  82. a_4_3·b_3_7 + b_2_3·b_4_10·a_1_1 + b_2_1·b_2_3·a_3_3 + b_2_12·a_3_3 + a_2_0·b_2_3·b_3_7
  83. b_4_10·b_3_5 + b_4_10·b_3_4 + b_2_1·b_5_16 + b_2_1·b_2_3·b_3_7 + b_2_12·b_3_6
       + b_2_12·b_3_5 + b_4_10·a_3_2 + b_2_1·b_2_3·a_3_3 + b_2_1·b_2_2·a_3_3 + b_2_12·a_3_3
       + b_2_12·a_3_2 + b_2_1·c_4_12·a_1_1 + b_2_1·c_4_11·a_1_1
  84. b_4_10·b_3_5 + b_2_3·b_5_16 + b_2_32·b_3_7 + b_2_12·b_3_7 + b_2_12·b_3_6
       + b_2_12·b_3_5 + b_2_12·b_3_4 + b_2_32·a_3_3 + b_2_2·b_2_3·a_3_3 + b_2_3·c_4_12·a_1_1
       + b_2_3·c_4_11·a_1_1
  85. b_4_10·b_3_6 + b_4_10·b_3_4 + b_2_2·b_5_16 + b_2_2·b_2_3·b_3_7 + b_2_12·b_3_5
       + b_4_10·a_3_3 + b_2_2·b_2_3·a_3_3 + b_2_1·b_2_3·a_3_3 + b_2_1·b_2_2·a_3_3
       + b_2_3·c_4_12·a_1_1 + b_2_1·c_4_11·a_1_1
  86. a_2_0·b_5_16 + a_2_0·b_2_3·b_3_7
  87. b_4_10·b_3_6 + b_4_10·b_3_4 + b_2_1·b_5_17 + b_2_1·b_2_2·b_3_7 + b_4_10·a_3_3
       + b_2_1·b_2_3·a_3_3 + b_2_12·a_3_2 + b_2_3·c_4_12·a_1_1 + b_2_1·c_4_12·a_1_1
  88. b_4_10·b_3_7 + b_4_10·b_3_6 + b_4_10·b_3_4 + b_2_3·b_5_17 + b_2_2·b_2_3·b_3_7
       + a_4_3·b_3_7 + b_2_32·a_3_3 + b_2_1·b_2_3·a_3_3 + b_2_1·b_2_2·a_3_3
       + b_2_3·c_4_12·a_1_1 + b_2_1·c_4_11·a_1_1
  89. b_4_10·b_3_6 + b_4_10·b_3_5 + b_4_10·b_3_4 + b_2_2·b_5_17 + b_2_1·b_2_3·b_3_7
       + b_2_1·b_2_2·b_3_7 + b_4_10·a_3_3 + b_2_2·b_2_3·a_3_3 + b_2_12·a_3_3
       + b_2_3·c_4_12·a_1_1
  90. a_4_3·b_3_7 + b_2_1·b_2_3·a_3_3 + b_2_12·a_3_3 + a_2_0·b_5_17 + a_2_0·b_2_3·b_3_7
  91. a_6_7·a_1_1
  92. a_6_7·a_1_0
  93. a_4_32
  94. a_4_3·a_4_4
  95. a_4_42
  96. b_4_102 + b_2_32·b_4_10 + b_2_2·b_2_33 + b_2_1·b_2_3·b_4_10 + b_2_1·b_2_2·b_2_32
       + b_2_12·b_2_32 + b_2_12·b_2_2·b_2_3 + b_2_13·b_2_3 + b_2_13·b_2_2 + b_2_14
       + b_2_32·a_4_4 + b_2_1·b_2_3·a_4_4 + a_2_0·b_2_33 + b_2_32·c_4_12 + b_2_12·c_4_11
  97. a_3_2·b_5_16 + a_4_3·b_4_10 + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_3 + b_2_12·a_4_3
       + a_2_0·b_2_3·c_4_12 + c_4_12·a_1_0·a_3_3 + c_4_12·a_1_0·a_3_2 + c_4_11·a_1_0·a_3_3
  98. b_3_4·b_5_16 + b_2_2·b_2_33 + b_2_1·b_2_3·b_4_10 + b_2_12·b_4_10 + b_2_13·b_2_3
       + b_2_13·b_2_2 + b_2_14 + a_4_3·b_4_10 + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_3·a_4_4
       + b_2_1·b_2_2·a_4_4 + b_2_1·b_2_2·a_4_3 + b_2_12·a_4_3 + a_2_0·b_2_3·c_4_12
       + c_4_12·a_1_0·a_3_3 + c_4_12·a_1_0·a_3_2 + c_4_11·a_1_0·a_3_3
  99. b_3_5·b_5_16 + b_2_2·b_2_33 + b_2_1·b_2_3·b_4_10 + b_2_1·b_2_2·b_2_32
       + b_2_12·b_2_2·b_2_3 + b_2_13·b_2_3 + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4
       + b_2_1·b_2_2·a_4_3 + b_2_12·a_4_4 + b_2_12·a_4_3
  100. b_3_6·b_5_16 + b_2_2·b_2_33 + b_2_1·b_2_3·b_4_10 + b_2_1·b_2_33 + b_2_1·b_2_2·b_4_10
       + b_2_1·b_2_2·b_2_32 + b_2_12·b_4_10 + b_2_12·b_2_32 + b_2_12·b_2_2·b_2_3
       + b_2_13·b_2_2 + b_2_14 + a_3_3·b_5_16 + a_4_3·b_4_10 + b_2_2·b_2_3·a_4_4
       + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_3 + b_2_12·a_4_4 + a_2_0·b_2_3·b_4_10
       + a_2_0·b_2_33 + a_2_0·b_2_3·c_4_12 + c_4_12·a_1_0·a_3_3 + c_4_11·a_1_0·a_3_3
       + c_4_11·a_1_0·a_3_2
  101. b_3_7·b_5_16 + b_2_34 + b_2_2·b_2_3·b_4_10 + b_2_2·b_2_33 + b_2_1·b_2_2·b_4_10
       + b_2_12·b_2_2·b_2_3 + b_2_13·b_2_3 + a_3_3·b_5_16 + b_2_2·b_2_3·a_4_4
       + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4 + a_2_0·b_2_33 + a_2_0·b_2_3·c_4_12
       + a_2_0·b_2_3·c_4_11 + c_4_12·a_1_0·a_3_2 + c_4_11·a_1_0·a_3_2
  102. a_3_3·b_5_16 + a_3_2·b_5_17 + b_2_32·a_4_4 + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_3·a_4_4
       + b_2_1·b_2_2·a_4_4 + b_2_12·a_4_4 + b_2_12·a_4_3 + a_2_0·b_2_33 + c_4_12·a_1_0·a_3_3
       + c_4_11·a_1_0·a_3_3 + c_4_11·a_1_0·a_3_2
  103. a_3_3·b_5_17 + a_3_3·b_5_16 + a_4_4·b_4_10 + a_4_3·b_4_10 + b_2_32·a_4_4
       + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4 + b_2_12·a_4_4 + b_2_12·a_4_3
       + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_33 + a_2_0·b_2_3·c_4_12 + c_4_11·a_1_0·a_3_3
       + c_4_11·a_1_0·a_3_2
  104. b_3_4·b_5_17 + b_2_2·b_2_3·b_4_10 + b_2_1·b_2_33 + b_2_1·b_2_2·b_4_10
       + b_2_1·b_2_2·b_2_32 + b_2_13·b_2_3 + b_2_13·b_2_2 + a_3_3·b_5_16 + b_2_32·a_4_4
       + b_2_12·a_4_3 + a_2_0·b_2_33 + c_4_12·a_1_0·a_3_3 + c_4_11·a_1_0·a_3_3
       + c_4_11·a_1_0·a_3_2
  105. b_3_5·b_5_17 + b_2_2·b_2_3·b_4_10 + b_2_1·b_2_33 + b_2_1·b_2_2·b_2_32
       + b_2_12·b_2_32 + b_2_12·b_2_2·b_2_3 + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_3·a_4_4
  106. b_3_6·b_5_17 + b_2_2·b_2_3·b_4_10 + b_2_1·b_2_3·b_4_10 + b_2_1·b_2_33
       + b_2_12·b_2_32 + a_4_4·b_4_10 + a_4_3·b_4_10 + b_2_1·b_2_2·a_4_4 + b_2_12·a_4_4
       + a_2_0·b_2_3·b_4_10 + c_4_12·a_1_0·a_3_3 + c_4_12·a_1_0·a_3_2
  107. b_3_7·b_5_17 + b_2_32·b_4_10 + b_2_2·b_2_3·b_4_10 + b_2_2·b_2_33 + b_2_1·b_2_3·b_4_10
       + b_2_1·b_2_33 + b_2_1·b_2_2·b_4_10 + b_2_1·b_2_2·b_2_32 + b_2_12·b_2_2·b_2_3
       + b_2_13·b_2_3 + b_2_13·b_2_2 + a_3_3·b_5_16 + a_4_4·b_4_10 + a_4_3·b_4_10
       + b_2_1·b_2_2·a_4_4 + b_2_12·a_4_4 + b_2_12·a_4_3 + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_33
       + a_2_0·b_2_3·c_4_12 + c_4_12·a_1_0·a_3_2 + c_4_11·a_1_0·a_3_3
  108. a_4_3·b_4_10 + b_2_1·a_6_7 + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4 + b_2_12·a_4_4
       + b_2_12·a_4_3 + a_2_0·b_2_3·c_4_12
  109. a_4_4·b_4_10 + a_4_3·b_4_10 + b_2_3·a_6_7 + b_2_32·a_4_4 + b_2_2·b_2_3·a_4_4
       + b_2_1·b_2_3·a_4_4 + b_2_12·a_4_4 + b_2_12·a_4_3 + a_2_0·b_2_3·b_4_10 + a_2_0·b_2_33
       + c_4_11·a_1_0·a_3_2
  110. a_3_3·b_5_16 + b_2_32·a_4_4 + b_2_2·a_6_7 + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_3
       + b_2_12·a_4_4 + b_2_12·a_4_3 + a_2_0·b_2_33 + c_4_12·a_1_0·a_3_2
       + c_4_11·a_1_0·a_3_3 + c_4_11·a_1_0·a_3_2
  111. a_2_0·a_6_7
  112. a_4_4·b_5_16 + b_2_33·a_3_3 + b_2_2·b_4_10·a_3_3 + b_2_1·b_4_10·a_3_3
       + b_2_1·b_4_10·a_3_2 + b_2_12·b_2_2·a_3_3 + b_2_13·a_3_2 + a_2_0·b_2_32·b_3_7
  113. a_4_3·b_5_17 + b_2_1·b_4_10·a_3_3 + b_2_1·b_2_2·b_2_3·a_3_3 + b_2_12·b_2_2·a_3_3
       + a_2_0·c_4_12·b_3_7
  114. a_4_3·b_5_16 + b_2_1·b_4_10·a_3_2 + b_2_1·b_2_32·a_3_3 + b_2_12·b_2_3·a_3_3
       + b_2_13·a_3_3 + b_2_13·a_3_2 + a_2_0·b_2_3·b_5_17 + a_2_0·b_2_32·b_3_7
  115. b_4_10·b_5_16 + b_2_32·b_5_17 + b_2_2·b_2_3·b_5_17 + b_2_2·b_2_32·b_3_7
       + b_2_1·b_2_2·b_5_17 + b_2_1·b_2_2·b_2_3·b_3_7 + b_2_12·b_5_16 + b_2_13·b_3_6
       + b_2_13·b_3_4 + a_4_3·b_5_16 + b_2_33·a_3_3 + b_2_2·b_4_10·a_3_3 + b_2_1·b_4_10·a_3_2
       + b_2_1·b_2_32·a_3_3 + b_2_12·b_2_2·a_3_3 + b_2_13·a_3_2 + b_2_2·c_4_12·b_3_7
       + b_2_1·c_4_12·b_3_7 + b_2_1·c_4_12·b_3_5 + b_2_1·c_4_11·b_3_5 + b_2_1·c_4_11·b_3_4
       + b_4_10·c_4_12·a_1_1 + b_4_10·c_4_11·a_1_1 + b_2_2·c_4_12·a_3_3 + b_2_1·c_4_12·a_3_3
       + b_2_1·c_4_11·a_3_2
  116. a_4_4·b_5_17 + a_4_3·b_5_16 + b_2_3·b_4_10·a_3_3 + b_2_2·b_2_32·a_3_3
       + b_2_1·b_4_10·a_3_3 + b_2_1·b_4_10·a_3_2 + b_2_1·b_2_32·a_3_3 + b_2_12·b_2_3·a_3_3
       + b_2_12·b_2_2·a_3_3 + b_2_13·a_3_3 + b_2_13·a_3_2 + a_2_0·b_2_32·b_3_7
  117. b_4_10·b_5_17 + b_4_10·b_5_16 + b_2_2·b_2_32·b_3_7 + b_2_1·b_2_3·b_5_17
       + b_2_1·b_2_32·b_3_7 + b_2_1·b_2_2·b_2_3·b_3_7 + b_2_12·b_5_16 + b_2_3·b_4_10·a_3_3
       + b_2_33·a_3_3 + b_2_1·b_4_10·a_3_2 + b_2_1·b_2_32·a_3_3 + b_2_12·b_2_3·a_3_3
       + b_2_12·b_2_2·a_3_3 + b_2_3·c_4_12·b_3_7 + b_2_2·c_4_12·b_3_7 + b_2_1·c_4_12·b_3_6
       + b_2_1·c_4_12·b_3_5 + b_2_1·c_4_12·b_3_4 + b_2_1·c_4_11·b_3_6 + b_2_1·c_4_11·b_3_5
       + b_4_10·c_4_12·a_1_1 + b_4_10·c_4_11·a_1_1 + b_2_3·c_4_12·a_3_3 + b_2_2·c_4_12·a_3_3
       + b_2_1·c_4_12·a_3_3 + b_2_1·c_4_11·a_3_3 + b_2_1·c_4_11·a_3_2 + a_2_0·c_4_12·b_3_7
  118. a_6_7·a_3_2
  119. a_6_7·a_3_3
  120. a_6_7·b_3_4 + b_2_2·b_4_10·a_3_3 + b_2_2·b_2_32·a_3_3 + b_2_1·b_4_10·a_3_3
       + b_2_1·b_4_10·a_3_2 + b_2_1·b_2_2·b_2_3·a_3_3 + b_2_12·b_2_3·a_3_3 + b_2_13·a_3_3
  121. a_6_7·b_3_5 + b_2_2·b_4_10·a_3_3 + b_2_2·b_2_32·a_3_3 + b_2_1·b_4_10·a_3_3
       + b_2_12·b_2_3·a_3_3
  122. a_6_7·b_3_6 + b_2_2·b_4_10·a_3_3 + b_2_2·b_2_32·a_3_3 + b_2_1·b_4_10·a_3_2
       + b_2_1·b_2_32·a_3_3 + b_2_12·b_2_3·a_3_3 + b_2_12·b_2_2·a_3_3 + b_2_13·a_3_3
  123. a_6_7·b_3_7 + b_2_3·b_4_10·a_3_3 + b_2_33·a_3_3 + b_2_2·b_2_32·a_3_3
       + b_2_1·b_4_10·a_3_3 + b_2_1·b_2_32·a_3_3 + b_2_12·b_2_2·a_3_3 + a_2_0·c_4_12·b_3_7
  124. b_5_162 + b_2_35 + b_2_2·b_2_34 + b_2_1·b_2_32·b_4_10 + b_2_1·b_2_2·b_2_33
       + b_2_12·b_2_3·b_4_10 + b_2_12·b_2_33 + b_2_13·b_2_32 + b_2_13·b_2_2·b_2_3
       + b_2_1·b_2_32·a_4_4 + b_2_12·b_2_3·a_4_4 + a_2_0·b_2_34 + b_2_1·b_2_32·c_4_12
       + b_2_13·c_4_11
  125. b_5_172 + b_2_33·b_4_10 + b_2_2·b_2_32·b_4_10 + b_2_2·b_2_34
       + b_2_1·b_2_32·b_4_10 + b_2_1·b_2_2·b_2_3·b_4_10 + b_2_12·b_2_33 + b_2_13·b_2_32
       + b_2_13·b_2_2·b_2_3 + b_2_14·b_2_3 + b_2_14·b_2_2 + b_2_33·a_4_4
       + b_2_2·b_2_32·a_4_4 + b_2_1·b_2_32·a_4_4 + b_2_1·b_2_2·b_2_3·a_4_4
       + a_2_0·b_2_32·b_4_10 + a_2_0·b_2_34 + b_2_33·c_4_12 + b_2_2·b_2_32·c_4_12
       + b_2_12·b_2_3·c_4_11 + b_2_12·b_2_2·c_4_11 + a_2_0·b_2_32·c_4_12
  126. a_4_3·a_6_7
  127. b_5_16·b_5_17 + b_2_33·b_4_10 + b_2_2·b_2_34 + b_2_1·b_2_2·b_2_3·b_4_10
       + b_2_12·b_2_33 + b_2_12·b_2_2·b_4_10 + b_2_14·b_2_3 + b_2_33·a_4_4
       + b_2_2·b_2_32·a_4_4 + b_2_1·b_2_2·b_2_3·a_4_4 + b_2_12·a_6_7 + b_2_12·b_2_3·a_4_4
       + b_2_12·b_2_2·a_4_4 + b_2_13·a_4_3 + b_2_2·b_2_32·c_4_12 + b_2_12·b_2_2·c_4_11
       + a_2_0·b_4_10·c_4_12 + a_2_0·b_4_10·c_4_11
  128. a_4_4·a_6_7
  129. b_4_10·a_6_7 + b_2_2·b_2_3·a_6_7 + b_2_1·b_2_3·a_6_7 + b_2_1·b_2_2·a_6_7
       + b_2_12·b_2_3·a_4_4 + b_2_13·a_4_4 + b_2_3·a_4_4·c_4_12 + b_2_1·a_4_4·c_4_12
       + b_2_1·a_4_3·c_4_12 + b_2_1·a_4_3·c_4_11 + a_2_0·b_4_10·c_4_12 + a_2_0·b_2_32·c_4_12
  130. a_6_7·b_5_17 + b_2_1·b_2_3·b_4_10·a_3_3 + b_2_1·b_2_33·a_3_3
       + b_2_1·b_2_2·b_2_32·a_3_3 + b_2_12·b_2_32·a_3_3 + b_2_12·b_2_2·b_2_3·a_3_3
       + b_2_14·a_3_3 + b_2_32·c_4_12·a_3_3 + b_2_12·c_4_11·a_3_3 + a_2_0·c_4_12·b_5_17
  131. a_6_7·b_5_16 + b_2_32·b_4_10·a_3_3 + b_2_34·a_3_3 + b_2_2·b_2_33·a_3_3
       + b_2_1·b_2_2·b_4_10·a_3_3 + b_2_12·b_4_10·a_3_3 + b_2_12·b_4_10·a_3_2
       + b_2_14·a_3_3 + b_2_14·a_3_2 + b_2_2·b_2_3·c_4_12·a_3_3 + b_2_1·b_2_3·c_4_12·a_3_3
       + b_2_12·c_4_11·a_3_2 + a_2_0·b_2_3·c_4_12·b_3_7
  132. a_6_72

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 12.
  • 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_11, a Duflot regular element of degree 4
    2. c_4_12, a Duflot regular element of degree 4
    3. b_2_3 + b_2_1, an element of degree 2
    4. b_3_5, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 9].
  • 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_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_2 → 0, an element of degree 3
  8. a_3_3 → 0, an element of degree 3
  9. b_3_4 → 0, an element of degree 3
  10. b_3_5 → 0, an element of degree 3
  11. b_3_6 → 0, an element of degree 3
  12. b_3_7 → 0, an element of degree 3
  13. a_4_3 → 0, an element of degree 4
  14. a_4_4 → 0, an element of degree 4
  15. b_4_10 → 0, an element of degree 4
  16. c_4_11 → c_1_14 + c_1_04, an element of degree 4
  17. c_4_12 → c_1_04, an element of degree 4
  18. b_5_16 → 0, an element of degree 5
  19. b_5_17 → 0, an element of degree 5
  20. a_6_7 → 0, an element of degree 6

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. 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_2·c_1_3 + c_1_22, an element of degree 2
  6. b_2_3 → c_1_32 + c_1_2·c_1_3, an element of degree 2
  7. a_3_2 → 0, an element of degree 3
  8. a_3_3 → 0, an element of degree 3
  9. b_3_4 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_23, an element of degree 3
  10. b_3_5 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  11. b_3_6 → c_1_2·c_1_32, an element of degree 3
  12. b_3_7 → c_1_33 + c_1_23, an element of degree 3
  13. a_4_3 → 0, an element of degree 4
  14. a_4_4 → 0, an element of degree 4
  15. b_4_10 → c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_23
       + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
  16. c_4_11 → c_1_24 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32
       + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22
       + c_1_04, an element of degree 4
  17. c_4_12 → c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22
       + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
  18. b_5_16 → c_1_35 + c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_23·c_1_32 + c_1_25
       + c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_0·c_1_24 + c_1_02·c_1_2·c_1_32
       + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_23, an element of degree 5
  19. b_5_17 → c_1_1·c_1_23·c_1_3 + c_1_1·c_1_24 + c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23
       + c_1_0·c_1_23·c_1_3 + c_1_0·c_1_24 + c_1_02·c_1_33 + c_1_02·c_1_23, an element of degree 5
  20. a_6_7 → 0, an element of degree 6

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