Cohomology of group number 33 of order 64

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

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 2 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 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1) · (t6  +  t3  +  1)
    (t  +  1) · (t  −  1)3 · (t2  +  1) · (t4  +  1)
  • The a-invariants are -∞,-4,-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 64

Ring generators

The cohomology ring has 19 minimal generators of maximal degree 9:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. b_2_1, an element of degree 2
  4. b_2_2, an element of degree 2
  5. b_3_2, an element of degree 3
  6. b_3_3, an element of degree 3
  7. b_3_4, an element of degree 3
  8. b_4_4, an element of degree 4
  9. b_4_6, an element of degree 4
  10. a_5_3, a nilpotent element of degree 5
  11. b_5_6, an element of degree 5
  12. b_5_7, an element of degree 5
  13. b_6_10, an element of degree 6
  14. b_6_11, an element of degree 6
  15. b_7_13, an element of degree 7
  16. b_7_14, an element of degree 7
  17. b_8_14, an element of degree 8
  18. c_8_18, a Duflot regular element of degree 8
  19. b_9_22, an element of degree 9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Ring relations

There are 134 minimal relations of maximal degree 18:

  1. a_1_02
  2. a_1_0·a_1_1
  3. a_1_13
  4. b_2_1·a_1_1
  5. b_2_2·a_1_0
  6. b_2_1·b_2_2
  7. a_1_0·b_3_2
  8. a_1_1·b_3_3 + a_1_1·b_3_2
  9. a_1_0·b_3_3
  10. a_1_0·b_3_4 + b_2_2·a_1_12
  11. b_2_1·b_3_2
  12. a_1_12·b_3_2
  13. b_2_2·b_3_3 + b_2_2·b_3_2 + b_2_22·a_1_1
  14. b_2_1·b_3_4 + b_2_1·b_3_3
  15. a_1_12·b_3_4
  16. b_4_4·a_1_1
  17. b_4_4·a_1_0
  18. b_4_6·a_1_0
  19. b_3_2·b_3_3 + b_3_22 + b_2_2·a_1_1·b_3_2
  20. b_3_3·b_3_4 + b_3_32 + b_3_2·b_3_4 + b_2_23 + b_2_2·a_1_1·b_3_4 + b_2_2·a_1_1·b_3_2
  21. b_2_2·b_4_4
  22. b_3_32 + b_3_22 + b_2_1·b_4_6
  23. b_3_22 + b_2_23 + b_2_2·a_1_1·b_3_2 + b_4_6·a_1_12
  24. b_3_22 + b_2_23 + b_2_2·a_1_1·b_3_2 + a_1_1·a_5_3
  25. a_1_0·a_5_3
  26. b_3_22 + b_2_23 + a_1_1·b_5_6 + b_2_2·a_1_1·b_3_2
  27. a_1_0·b_5_6
  28. b_3_42 + b_3_32 + b_3_22 + b_2_2·b_4_6 + a_1_1·b_5_7
  29. a_1_0·b_5_7
  30. b_4_4·b_3_2
  31. b_4_4·b_3_4 + b_4_4·b_3_3
  32. a_1_1·b_3_2·b_3_4 + b_2_2·a_5_3 + b_2_2·b_4_6·a_1_1
  33. b_4_4·b_3_3 + b_2_1·b_5_6 + b_2_1·a_5_3
  34. b_2_2·b_5_6 + b_2_2·b_4_6·a_1_1 + b_2_23·a_1_1
  35. b_2_1·b_5_7
  36. a_1_12·b_5_7
  37. a_1_1·b_3_2·b_3_4 + b_6_10·a_1_1 + b_2_23·a_1_1
  38. b_6_10·a_1_0 + b_2_1·a_5_3
  39. b_4_6·b_3_2 + b_2_2·b_5_7 + a_1_1·b_3_2·b_3_4 + b_6_11·a_1_1 + b_2_2·b_4_6·a_1_1
  40. b_6_11·a_1_0
  41. b_4_42 + b_2_12·b_4_6
  42. b_3_3·a_5_3 + b_3_2·a_5_3
  43. b_3_2·b_5_6 + b_3_2·a_5_3 + b_2_22·a_1_1·b_3_4 + b_2_22·a_1_1·b_3_2
  44. b_3_3·b_5_6 + b_4_4·b_4_6 + b_3_2·a_5_3 + b_2_22·a_1_1·b_3_4 + b_2_22·a_1_1·b_3_2
  45. b_3_4·b_5_6 + b_4_4·b_4_6 + b_3_4·a_5_3 + b_3_2·a_5_3
  46. b_3_4·a_5_3 + b_4_6·a_1_1·b_3_4 + b_2_2·a_1_1·b_5_7
  47. b_3_3·b_5_7 + b_3_2·b_5_7 + b_3_2·a_5_3 + b_2_22·a_1_1·b_3_4
  48. b_2_2·b_3_2·b_3_4 + b_2_2·b_6_10 + b_2_24 + b_3_4·a_5_3 + b_4_6·a_1_1·b_3_4
  49. b_4_4·b_4_6 + b_2_1·b_6_11 + b_2_12·b_4_4
  50. b_3_4·a_5_3 + b_3_2·a_5_3 + b_4_6·a_1_1·b_3_4 + b_2_22·a_1_1·b_3_4 + b_6_11·a_1_12
  51. b_3_2·b_5_7 + b_2_22·b_4_6 + b_3_4·a_5_3 + b_3_2·a_5_3 + a_1_1·b_7_13
  52. a_1_0·b_7_13
  53. b_3_4·a_5_3 + a_1_1·b_7_14 + b_4_6·a_1_1·b_3_4 + b_2_22·a_1_1·b_3_2
  54. a_1_0·b_7_14
  55. b_4_4·a_5_3
  56. b_4_4·b_5_6 + b_2_1·b_4_6·b_3_3
  57. a_1_1·b_3_4·b_5_7 + b_4_6·a_5_3 + b_4_62·a_1_1
  58. b_4_4·b_5_7
  59. b_6_10·b_3_2 + b_2_23·b_3_4 + b_2_23·b_3_2 + b_2_22·a_5_3
  60. b_6_10·b_3_4 + b_6_10·b_3_3 + b_2_22·b_5_7 + b_2_23·b_3_2 + b_4_6·a_5_3 + b_4_62·a_1_1
       + b_2_2·b_6_11·a_1_1 + b_2_22·a_5_3 + b_2_24·a_1_1
  61. b_6_11·b_3_3 + b_6_11·b_3_2 + b_6_10·b_3_4 + b_6_10·b_3_3 + b_4_6·b_5_6 + b_2_22·b_5_7
       + b_2_23·b_3_2 + b_2_12·b_5_6 + b_4_6·a_5_3 + b_2_22·a_5_3 + b_2_22·b_4_6·a_1_1
       + b_2_24·a_1_1 + b_2_12·a_5_3
  62. b_2_1·b_7_13 + b_2_1·b_4_6·b_3_3 + b_2_13·b_3_3
  63. b_6_11·b_3_2 + b_6_10·b_3_4 + b_6_10·b_3_3 + b_2_2·b_7_13 + b_2_2·b_4_6·b_3_4
       + b_2_22·b_5_7 + b_2_23·b_3_2 + b_4_6·a_5_3 + b_2_22·b_4_6·a_1_1
  64. a_1_12·b_7_13
  65. b_6_10·b_3_3 + b_2_23·b_3_4 + b_2_23·b_3_2 + b_2_1·b_7_14 + b_2_12·b_5_6
       + b_2_22·b_4_6·a_1_1 + b_2_24·a_1_1 + b_2_12·a_5_3
  66. b_6_10·b_3_4 + b_6_10·b_3_3 + b_2_2·b_7_14 + b_2_22·a_5_3 + b_2_24·a_1_1
  67. b_8_14·a_1_1 + b_4_6·a_5_3 + b_4_62·a_1_1 + b_2_22·a_5_3 + b_2_22·b_4_6·a_1_1
       + b_2_24·a_1_1
  68. b_8_14·a_1_0
  69. a_5_32 + b_4_62·a_1_12
  70. a_5_3·b_5_6 + b_4_62·a_1_12
  71. b_5_62 + b_2_1·b_4_62 + b_4_62·a_1_12
  72. a_5_3·b_5_7 + b_4_6·a_1_1·b_5_7 + b_2_2·b_4_6·a_1_1·b_3_4
  73. b_5_6·b_5_7 + a_5_3·b_5_7 + b_2_2·b_4_6·a_1_1·b_3_4 + b_2_22·a_1_1·b_5_7
       + b_4_62·a_1_12
  74. b_4_4·b_6_11 + b_2_1·b_4_62 + b_2_13·b_4_6
  75. b_3_2·b_7_13 + b_2_2·b_3_4·b_5_7 + b_2_22·b_6_11 + a_5_3·b_5_7 + b_6_11·a_1_1·b_3_4
       + b_2_22·a_1_1·b_5_7 + b_2_23·a_1_1·b_3_4 + b_2_23·a_1_1·b_3_2 + b_4_62·a_1_12
  76. b_3_3·b_7_13 + b_2_2·b_3_4·b_5_7 + b_2_22·b_6_11 + b_2_1·b_4_62 + b_2_13·b_4_6
       + a_5_3·b_5_7 + b_6_11·a_1_1·b_3_4 + b_2_2·a_1_1·b_7_13 + b_2_22·a_1_1·b_5_7
       + b_2_23·a_1_1·b_3_4 + b_2_23·a_1_1·b_3_2 + b_4_62·a_1_12
  77. b_3_2·b_7_14 + b_2_23·b_4_6 + b_2_25 + b_2_2·b_4_6·a_1_1·b_3_4 + b_2_23·a_1_1·b_3_4
       + b_2_23·a_1_1·b_3_2
  78. b_3_3·b_7_14 + b_4_6·b_6_10 + b_2_2·b_3_4·b_5_7 + b_2_25 + b_2_12·b_6_11
       + b_2_13·b_4_4 + a_5_3·b_5_7 + b_6_11·a_1_1·b_3_4 + b_2_2·b_4_6·a_1_1·b_3_4
       + b_2_23·a_1_1·b_3_4
  79. b_3_4·b_7_14 + b_4_6·b_6_10 + b_2_22·b_6_10 + b_2_23·b_4_6 + b_2_25 + b_2_12·b_6_11
       + b_2_13·b_4_4 + b_2_2·b_4_6·a_1_1·b_3_4
  80. b_5_72 + b_2_2·b_4_62 + a_5_3·b_5_7 + b_2_2·b_4_6·a_1_1·b_3_4 + b_4_62·a_1_12
       + c_8_18·a_1_12
  81. b_4_4·b_6_10 + b_2_1·b_8_14 + b_2_13·b_4_4
  82. b_2_2·b_3_4·b_5_7 + b_2_2·b_8_14 + b_2_22·b_6_10 + b_2_2·a_1_1·b_7_13
       + b_2_22·a_1_1·b_5_7 + b_2_23·a_1_1·b_3_2
  83. a_5_3·b_5_7 + a_1_1·b_9_22 + b_2_22·a_1_1·b_5_7 + b_2_23·a_1_1·b_3_2
       + b_4_62·a_1_12
  84. a_1_0·b_9_22
  85. b_6_10·b_5_7 + b_2_22·b_4_6·b_3_4 + b_2_23·b_5_7 + b_6_11·a_5_3 + b_4_6·b_6_11·a_1_1
       + b_2_2·b_4_62·a_1_1 + b_2_23·b_4_6·a_1_1
  86. b_6_11·b_5_6 + b_4_62·b_3_3 + b_2_2·b_4_6·b_5_7 + b_2_12·b_4_6·b_3_3
       + b_2_2·b_4_6·a_5_3 + b_2_2·b_4_62·a_1_1 + b_2_22·b_6_11·a_1_1
  87. b_6_10·b_5_7 + b_2_22·b_4_6·b_3_4 + b_2_23·b_5_7 + a_1_1·b_3_4·b_7_13
       + b_2_23·b_4_6·a_1_1
  88. b_4_4·b_7_13 + b_2_1·b_4_6·b_5_6 + b_2_13·b_5_6 + b_2_13·a_5_3
  89. b_6_10·b_5_6 + b_4_4·b_7_14 + b_2_12·b_4_6·b_3_3 + b_6_10·a_5_3 + b_2_23·b_4_6·a_1_1
       + b_2_25·a_1_1
  90. b_6_10·a_5_3 + b_2_2·b_4_6·a_5_3 + b_2_2·b_4_62·a_1_1 + b_2_23·a_5_3
       + b_2_23·b_4_6·a_1_1 + b_2_1·c_8_18·a_1_0
  91. b_6_11·b_5_7 + b_4_6·b_7_13 + b_4_62·b_3_4 + b_2_12·b_4_6·b_3_3 + b_6_11·a_5_3
       + b_2_2·b_4_6·a_5_3 + b_2_2·b_4_62·a_1_1 + b_2_23·b_4_6·a_1_1 + b_2_2·c_8_18·a_1_1
  92. b_8_14·b_3_2 + b_6_10·b_5_7 + b_2_23·b_5_7 + b_2_24·b_3_4 + b_2_24·b_3_2
       + b_2_2·b_4_6·a_5_3 + b_2_22·b_6_11·a_1_1 + b_2_23·a_5_3 + b_2_23·b_4_6·a_1_1
       + b_2_25·a_1_1
  93. b_8_14·b_3_3 + b_6_10·b_5_7 + b_6_10·b_5_6 + b_2_23·b_5_7 + b_2_24·b_3_4 + b_2_24·b_3_2
       + b_2_13·b_5_6 + b_6_10·a_5_3 + b_2_2·b_4_62·a_1_1 + b_2_22·b_6_11·a_1_1
       + b_2_23·b_4_6·a_1_1 + b_2_25·a_1_1 + b_2_13·a_5_3
  94. b_8_14·b_3_4 + b_6_10·b_5_7 + b_6_10·b_5_6 + b_2_2·b_4_6·b_5_7 + b_2_22·b_4_6·b_3_4
       + b_2_24·b_3_4 + b_2_13·b_5_6 + b_6_10·a_5_3 + b_2_2·b_4_62·a_1_1
       + b_2_22·b_6_11·a_1_1 + b_2_25·a_1_1 + b_2_13·a_5_3
  95. b_6_10·b_5_6 + b_2_1·b_9_22 + b_2_1·b_4_6·b_5_6 + b_2_12·b_7_14 + b_2_12·b_4_6·b_3_3
       + b_2_2·b_4_6·a_5_3 + b_2_2·b_4_62·a_1_1 + b_2_23·a_5_3 + b_2_25·a_1_1
  96. b_6_10·b_5_7 + b_2_2·b_9_22 + b_2_2·b_4_6·b_5_7 + b_2_24·b_3_2 + b_2_2·b_4_6·a_5_3
       + b_2_22·b_6_11·a_1_1 + b_2_23·b_4_6·a_1_1
  97. a_5_3·b_7_13 + b_4_6·a_1_1·b_7_13 + b_2_2·b_6_11·a_1_1·b_3_4 + b_2_2·b_4_6·a_1_1·b_5_7
  98. b_5_6·b_7_13 + b_2_1·b_4_6·b_6_11 + a_5_3·b_7_13 + b_2_2·b_6_11·a_1_1·b_3_4
       + b_2_2·b_4_6·a_1_1·b_5_7 + b_2_22·a_1_1·b_7_13
  99. a_5_3·b_7_14 + b_2_2·b_4_6·a_1_1·b_5_7 + b_2_22·b_4_6·a_1_1·b_3_4
       + b_2_23·a_1_1·b_5_7 + b_2_24·a_1_1·b_3_4
  100. b_5_7·b_7_14 + b_5_6·b_7_14 + b_6_10·b_6_11 + b_6_102 + b_2_2·b_3_4·b_7_13
       + b_2_23·b_6_11 + b_2_26 + b_2_14·b_4_4 + b_2_2·b_6_11·a_1_1·b_3_4
       + b_2_2·b_4_6·a_1_1·b_5_7 + b_2_22·a_1_1·b_7_13 + b_2_22·b_4_6·a_1_1·b_3_4
       + b_2_23·a_1_1·b_5_7 + b_2_24·a_1_1·b_3_2 + b_2_12·c_8_18
  101. b_6_112 + b_4_63 + b_2_2·b_4_6·b_6_11 + b_2_14·b_4_6 + b_4_62·a_1_1·b_3_4
       + b_2_2·b_6_11·a_1_1·b_3_4 + b_2_2·b_4_6·a_1_1·b_5_7 + b_2_22·b_4_6·a_1_1·b_3_4
       + b_2_22·c_8_18
  102. b_5_7·b_7_14 + b_2_22·b_4_62 + b_2_24·b_4_6 + a_5_3·b_7_13
       + b_2_2·b_6_11·a_1_1·b_3_4 + b_2_22·a_1_1·b_7_13 + b_2_22·b_4_6·a_1_1·b_3_4
       + b_2_24·a_1_1·b_3_4 + b_4_6·b_6_11·a_1_12 + b_2_2·c_8_18·a_1_12
  103. b_5_7·b_7_14 + b_5_7·b_7_13 + b_4_6·b_3_4·b_5_7 + b_2_2·b_4_6·b_6_11 + b_2_22·b_4_62
       + b_2_24·b_4_6 + a_5_3·b_7_13 + b_2_22·a_1_1·b_7_13 + b_2_23·a_1_1·b_5_7
       + b_2_24·a_1_1·b_3_4 + c_8_18·a_1_1·b_3_2
  104. b_5_7·b_7_14 + b_5_6·b_7_14 + b_6_10·b_6_11 + b_2_2·b_3_4·b_7_13 + b_2_23·b_6_11
       + b_2_24·b_4_6 + b_2_12·b_8_14 + b_2_12·b_4_62 + b_2_14·b_4_4
       + b_2_2·b_6_11·a_1_1·b_3_4 + b_2_2·b_4_6·a_1_1·b_5_7 + b_2_22·a_1_1·b_7_13
       + b_2_22·b_4_6·a_1_1·b_3_4 + b_2_23·a_1_1·b_5_7 + b_2_24·a_1_1·b_3_2
  105. b_5_6·b_7_14 + b_4_6·b_3_4·b_5_7 + b_4_6·b_8_14 + b_2_22·b_8_14 + b_2_23·b_6_10
       + b_2_24·b_4_6 + b_2_12·b_4_62 + b_2_13·b_6_11 + b_2_14·b_4_4 + a_5_3·b_7_13
       + b_2_22·a_1_1·b_7_13 + b_2_22·b_4_6·a_1_1·b_3_4 + b_2_23·a_1_1·b_5_7
       + b_4_6·b_6_11·a_1_12
  106. b_4_4·b_8_14 + b_2_1·b_4_6·b_6_10 + b_2_14·b_4_6
  107. b_3_2·b_9_22 + b_2_22·b_8_14 + b_2_22·b_4_62 + b_2_23·b_6_10 + b_2_24·b_4_6
       + b_2_26 + a_5_3·b_7_13 + b_4_62·a_1_1·b_3_4 + b_2_2·b_6_11·a_1_1·b_3_4
       + b_2_2·b_4_6·a_1_1·b_5_7 + b_2_22·a_1_1·b_7_13 + b_2_22·b_4_6·a_1_1·b_3_4
       + b_2_24·a_1_1·b_3_4
  108. b_5_6·b_7_14 + b_3_3·b_9_22 + b_2_22·b_8_14 + b_2_22·b_4_62 + b_2_23·b_6_10
       + b_2_24·b_4_6 + b_2_26 + b_2_1·b_4_6·b_6_11 + b_2_1·b_4_6·b_6_10 + a_5_3·b_7_13
       + b_4_62·a_1_1·b_3_4 + b_2_2·b_6_11·a_1_1·b_3_4 + b_2_2·b_4_6·a_1_1·b_5_7
       + b_2_22·a_1_1·b_7_13 + b_2_23·a_1_1·b_5_7 + b_2_24·a_1_1·b_3_4
       + b_4_6·b_6_11·a_1_12
  109. b_5_7·b_7_14 + b_5_6·b_7_14 + b_3_4·b_9_22 + b_4_6·b_3_4·b_5_7 + b_2_22·b_8_14
       + b_2_24·b_4_6 + b_2_26 + b_2_1·b_4_6·b_6_11 + b_2_1·b_4_6·b_6_10
       + b_4_62·a_1_1·b_3_4 + b_2_2·b_6_11·a_1_1·b_3_4 + b_2_22·b_4_6·a_1_1·b_3_4
       + b_2_24·a_1_1·b_3_4 + b_4_6·b_6_11·a_1_12
  110. b_6_10·b_7_13 + b_2_22·b_6_11·b_3_4 + b_2_22·b_4_6·b_5_7 + b_2_23·b_7_13
       + b_2_1·b_4_6·b_7_14 + b_2_12·b_4_6·b_5_6 + b_2_13·b_7_14 + b_2_14·b_5_6
       + b_2_22·b_4_6·a_5_3 + b_2_22·b_4_62·a_1_1 + b_2_24·a_5_3 + b_2_14·a_5_3
  111. b_6_11·b_7_13 + b_4_6·b_6_11·b_3_4 + b_4_62·b_5_7 + b_2_2·b_4_6·b_7_13
       + b_2_2·b_4_62·b_3_4 + b_2_12·b_4_6·b_5_6 + b_2_14·b_5_6 + b_4_63·a_1_1
       + b_2_22·b_4_62·a_1_1 + b_2_23·b_6_11·a_1_1 + b_2_24·b_4_6·a_1_1 + b_2_14·a_5_3
       + b_2_2·c_8_18·b_3_2 + b_2_22·c_8_18·a_1_1
  112. b_6_10·b_7_14 + b_2_23·b_4_6·b_3_4 + b_2_24·b_5_7 + b_2_25·b_3_4 + b_2_25·b_3_2
       + b_2_1·b_4_62·b_3_3 + b_2_14·b_5_6 + b_2_22·b_4_6·a_5_3 + b_2_22·b_4_62·a_1_1
       + b_2_23·b_6_11·a_1_1 + b_2_24·a_5_3 + b_2_24·b_4_6·a_1_1 + b_2_14·a_5_3
       + b_2_1·c_8_18·b_3_3
  113. b_8_14·a_5_3 + b_4_62·a_5_3 + b_4_63·a_1_1 + b_2_22·b_4_6·a_5_3 + b_2_24·a_5_3
       + b_2_24·b_4_6·a_1_1
  114. b_8_14·b_5_6 + b_6_10·b_7_13 + b_2_22·b_6_11·b_3_4 + b_2_22·b_4_6·b_5_7
       + b_2_23·b_7_13 + b_2_13·b_7_14 + b_2_13·b_4_6·b_3_3 + b_2_14·b_5_6 + b_4_62·a_5_3
       + b_4_63·a_1_1 + b_2_22·b_4_6·a_5_3 + b_2_22·b_4_62·a_1_1 + b_2_26·a_1_1
       + b_2_14·a_5_3
  115. b_8_14·b_5_7 + b_2_2·b_4_62·b_3_4 + b_2_23·b_4_6·b_3_4 + b_2_24·b_5_7
       + b_2_2·b_6_11·a_5_3
  116. b_6_11·b_7_14 + b_6_10·b_7_13 + b_4_6·b_9_22 + b_4_62·b_5_7 + b_4_62·b_5_6
       + b_2_2·b_4_6·b_7_13 + b_2_22·b_6_11·b_3_4 + b_2_23·b_4_6·b_3_4 + b_2_24·b_5_7
       + b_2_12·b_9_22 + b_2_14·b_5_6 + b_4_62·a_5_3 + b_4_63·a_1_1 + b_2_2·b_6_11·a_5_3
       + b_2_24·a_5_3 + b_2_26·a_1_1 + b_2_14·a_5_3 + b_2_12·c_8_18·a_1_0
  117. b_6_10·b_7_13 + b_4_4·b_9_22 + b_2_22·b_6_11·b_3_4 + b_2_22·b_4_6·b_5_7
       + b_2_23·b_7_13 + b_2_1·b_4_62·b_3_3 + b_2_12·b_9_22 + b_2_14·b_5_6
       + b_2_22·b_4_6·a_5_3 + b_2_22·b_4_62·a_1_1 + b_2_24·a_5_3 + b_2_14·a_5_3
       + b_2_12·c_8_18·a_1_0
  118. b_7_132 + b_2_22·b_4_6·b_6_11 + b_2_1·b_4_63 + b_2_15·b_4_6
       + b_2_2·b_4_6·a_1_1·b_7_13 + b_2_22·b_6_11·a_1_1·b_3_4 + b_2_22·b_4_6·a_1_1·b_5_7
       + b_2_23·b_4_6·a_1_1·b_3_4 + b_2_23·c_8_18 + b_2_2·c_8_18·a_1_1·b_3_2
       + b_4_6·c_8_18·a_1_12
  119. b_7_13·b_7_14 + b_4_62·b_6_10 + b_2_22·b_4_6·b_6_11 + b_2_23·b_8_14
       + b_2_23·b_4_62 + b_2_24·b_6_11 + b_2_24·b_6_10 + b_2_12·b_4_6·b_6_11
       + b_2_12·b_4_6·b_6_10 + b_4_6·b_6_11·a_1_1·b_3_4 + b_4_62·a_1_1·b_5_7
       + b_2_2·b_4_6·a_1_1·b_7_13 + b_2_23·a_1_1·b_7_13 + b_2_24·a_1_1·b_5_7
       + b_2_25·a_1_1·b_3_4
  120. b_7_13·b_7_14 + b_6_10·b_8_14 + b_4_62·b_6_10 + b_2_22·b_4_6·b_6_11 + b_2_24·b_6_11
       + b_2_25·b_4_6 + b_2_27 + b_2_13·b_8_14 + b_2_14·b_6_11 + b_2_15·b_4_6
       + b_4_6·b_6_11·a_1_1·b_3_4 + b_4_62·a_1_1·b_5_7 + b_2_22·b_6_11·a_1_1·b_3_4
       + b_2_23·a_1_1·b_7_13 + b_2_24·a_1_1·b_5_7 + b_2_1·b_4_4·c_8_18
  121. b_7_142 + b_2_23·b_4_62 + b_2_27 + b_2_1·b_4_6·b_8_14 + b_2_1·b_4_63
       + b_2_13·b_4_62 + b_2_22·b_4_6·a_1_1·b_5_7 + b_2_25·a_1_1·b_3_2
       + b_2_1·b_4_6·c_8_18
  122. b_7_13·b_7_14 + b_6_11·b_8_14 + b_4_6·b_3_4·b_7_13 + b_2_2·b_4_6·b_8_14 + b_2_2·b_4_63
       + b_2_22·b_3_4·b_7_13 + b_2_22·b_4_6·b_6_11 + b_2_23·b_4_62 + b_2_25·b_4_6
       + b_2_1·b_4_63 + b_2_12·b_4_6·b_6_11 + b_2_15·b_4_6 + b_2_22·b_4_6·a_1_1·b_5_7
       + b_2_23·a_1_1·b_7_13 + b_2_23·b_4_6·a_1_1·b_3_4 + b_2_24·a_1_1·b_5_7
       + b_2_25·a_1_1·b_3_2 + b_2_2·c_8_18·a_1_1·b_3_4 + b_2_2·c_8_18·a_1_1·b_3_2
  123. a_5_3·b_9_22 + b_4_62·a_1_1·b_5_7 + b_2_23·b_4_6·a_1_1·b_3_4 + b_2_24·a_1_1·b_5_7
       + b_2_25·a_1_1·b_3_4 + b_4_63·a_1_12
  124. b_7_142 + b_5_6·b_9_22 + b_6_11·b_8_14 + b_4_6·b_3_4·b_7_13 + b_2_2·b_4_63
       + b_2_22·b_3_4·b_7_13 + b_2_24·b_6_11 + b_2_27 + b_2_1·b_4_63
       + b_2_12·b_4_6·b_6_11 + b_2_12·b_4_6·b_6_10 + b_2_15·b_4_6 + b_4_62·a_1_1·b_5_7
       + b_2_22·b_6_11·a_1_1·b_3_4 + b_2_22·b_4_6·a_1_1·b_5_7 + b_2_23·a_1_1·b_7_13
       + b_2_23·b_4_6·a_1_1·b_3_4 + b_2_24·a_1_1·b_5_7 + b_2_25·a_1_1·b_3_4
       + b_2_1·b_4_6·c_8_18 + b_2_2·c_8_18·a_1_1·b_3_4 + b_2_2·c_8_18·a_1_1·b_3_2
  125. b_7_132 + b_5_7·b_9_22 + b_6_11·b_8_14 + b_4_6·b_3_4·b_7_13 + b_4_62·b_6_10
       + b_2_22·b_3_4·b_7_13 + b_2_22·b_4_6·b_6_11 + b_2_23·b_4_62 + b_2_24·b_6_11
       + b_2_25·b_4_6 + b_2_12·b_4_6·b_6_10 + b_2_22·b_4_6·a_1_1·b_5_7
       + b_2_24·a_1_1·b_5_7 + b_2_23·c_8_18 + b_2_2·c_8_18·a_1_1·b_3_4
  126. b_8_14·b_7_14 + b_2_22·b_4_62·b_3_4 + b_2_25·b_5_7 + b_2_26·b_3_4 + b_2_26·b_3_2
       + b_2_1·b_4_62·b_5_6 + b_2_13·b_9_22 + b_2_13·b_4_6·b_5_6 + b_2_14·b_7_14
       + b_2_14·b_4_6·b_3_3 + b_4_6·b_6_11·a_5_3 + b_4_62·b_6_11·a_1_1 + b_2_2·b_4_63·a_1_1
       + b_2_22·b_6_11·a_5_3 + b_2_23·b_4_62·a_1_1 + b_2_25·a_5_3 + b_2_27·a_1_1
       + b_2_1·c_8_18·b_5_6 + b_2_1·c_8_18·a_5_3 + b_2_13·c_8_18·a_1_0
  127. b_8_14·b_7_13 + b_6_11·b_9_22 + b_4_62·b_7_14 + b_4_62·b_7_13 + b_4_63·b_3_4
       + b_4_63·b_3_3 + b_2_2·b_4_62·b_5_7 + b_2_22·b_4_6·b_7_13 + b_2_22·b_4_62·b_3_4
       + b_2_23·b_6_11·b_3_4 + b_2_24·b_4_6·b_3_4 + b_2_12·b_4_6·b_7_14
       + b_2_12·b_4_62·b_3_3 + b_2_13·b_4_6·b_5_6 + b_2_14·b_4_6·b_3_3 + b_2_15·b_5_6
       + b_4_6·b_6_11·a_5_3 + b_2_2·b_4_62·a_5_3 + b_2_2·b_4_63·a_1_1 + b_2_22·b_6_11·a_5_3
       + b_2_22·b_4_6·b_6_11·a_1_1 + b_2_23·b_4_6·a_5_3 + b_2_23·b_4_62·a_1_1
       + b_2_27·a_1_1 + b_2_15·a_5_3 + b_2_2·b_4_6·c_8_18·a_1_1 + b_2_23·c_8_18·a_1_1
  128. b_8_14·b_7_14 + b_8_14·b_7_13 + b_6_10·b_9_22 + b_2_2·b_4_6·b_6_11·b_3_4
       + b_2_2·b_4_62·b_5_7 + b_2_23·b_6_11·b_3_4 + b_2_23·b_4_6·b_5_7 + b_2_24·b_7_13
       + b_2_12·b_4_62·b_3_3 + b_2_13·b_4_6·b_5_6 + b_2_14·b_4_6·b_3_3
       + b_2_2·b_4_63·a_1_1 + b_2_23·b_4_6·a_5_3 + b_2_25·a_5_3 + b_2_25·b_4_6·a_1_1
       + b_2_27·a_1_1 + b_2_12·c_8_18·b_3_3 + b_2_2·c_8_18·a_5_3 + b_2_2·b_4_6·c_8_18·a_1_1
       + b_2_23·c_8_18·a_1_1 + b_2_1·c_8_18·a_5_3
  129. b_8_14·b_7_14 + b_8_14·b_7_13 + b_2_2·b_4_6·b_6_11·b_3_4 + b_2_2·b_4_62·b_5_7
       + b_2_22·b_4_62·b_3_4 + b_2_23·b_6_11·b_3_4 + b_2_23·b_4_6·b_5_7 + b_2_24·b_7_13
       + b_2_25·b_5_7 + b_2_26·b_3_4 + b_2_26·b_3_2 + b_2_1·b_4_6·b_9_22
       + b_2_12·b_4_6·b_7_14 + b_2_12·b_4_62·b_3_3 + b_2_13·b_4_6·b_5_6 + b_2_15·b_5_6
       + b_4_6·b_6_11·a_5_3 + b_4_62·b_6_11·a_1_1 + b_2_2·b_4_62·a_5_3 + b_2_2·b_4_63·a_1_1
       + b_2_22·b_6_11·a_5_3 + b_2_22·b_4_6·b_6_11·a_1_1 + b_2_23·b_4_6·a_5_3
       + b_2_23·b_4_62·a_1_1 + b_2_24·b_6_11·a_1_1 + b_2_27·a_1_1 + b_2_15·a_5_3
       + b_2_1·c_8_18·b_5_6 + b_2_2·c_8_18·a_5_3 + b_2_2·b_4_6·c_8_18·a_1_1
       + b_2_23·c_8_18·a_1_1 + b_2_1·c_8_18·a_5_3
  130. b_8_142 + b_2_22·b_4_63 + b_2_26·b_4_6 + b_2_28 + b_2_12·b_4_6·b_8_14
       + b_2_12·b_4_63 + b_2_16·b_4_6 + b_2_12·b_4_6·c_8_18
  131. b_7_14·b_9_22 + b_4_62·b_3_4·b_5_7 + b_4_62·b_8_14 + b_2_22·b_4_63 + b_2_24·b_8_14
       + b_2_24·b_4_62 + b_2_25·b_6_10 + b_2_28 + b_2_1·b_4_62·b_6_11
       + b_2_1·b_4_62·b_6_10 + b_2_12·b_4_6·b_8_14 + b_2_13·b_4_6·b_6_11 + b_2_15·b_6_11
       + b_2_16·b_4_4 + b_2_2·b_4_6·b_6_11·a_1_1·b_3_4 + b_2_22·b_4_6·a_1_1·b_7_13
       + b_2_23·b_4_6·a_1_1·b_5_7 + b_2_24·a_1_1·b_7_13 + b_2_1·b_6_11·c_8_18
       + b_2_12·b_4_6·c_8_18 + b_2_12·b_4_4·c_8_18
  132. b_7_13·b_9_22 + b_4_62·b_8_14 + b_2_2·b_4_6·b_3_4·b_7_13 + b_2_2·b_4_62·b_6_11
       + b_2_23·b_4_6·b_6_11 + b_2_24·b_8_14 + b_2_24·b_4_62 + b_2_25·b_6_11
       + b_2_25·b_6_10 + b_2_1·b_4_62·b_6_11 + b_2_1·b_4_62·b_6_10 + b_2_12·b_4_6·b_8_14
       + b_2_12·b_4_63 + b_2_13·b_4_6·b_6_10 + b_2_14·b_4_62
       + b_2_2·b_4_6·b_6_11·a_1_1·b_3_4 + b_2_22·b_4_6·a_1_1·b_7_13
       + b_2_22·b_4_62·a_1_1·b_3_4 + b_2_24·a_1_1·b_7_13 + b_2_25·a_1_1·b_5_7
       + b_2_26·a_1_1·b_3_4 + b_2_2·c_8_18·a_1_1·b_5_7 + b_2_22·c_8_18·a_1_1·b_3_4
       + b_6_11·c_8_18·a_1_12
  133. b_8_14·b_9_22 + b_2_2·b_4_63·b_3_4 + b_2_22·b_4_62·b_5_7 + b_2_25·b_4_6·b_3_4
       + b_2_26·b_5_7 + b_2_27·b_3_4 + b_2_27·b_3_2 + b_2_1·b_4_62·b_7_14
       + b_2_1·b_4_63·b_3_3 + b_2_13·b_4_6·b_7_14 + b_2_13·b_4_62·b_3_3 + b_2_14·b_9_22
       + b_2_15·b_7_14 + b_2_15·b_4_6·b_3_3 + b_2_2·b_4_6·b_6_11·a_5_3
       + b_2_2·b_4_62·b_6_11·a_1_1 + b_2_22·b_4_63·a_1_1 + b_2_23·b_4_6·b_6_11·a_1_1
       + b_2_24·b_4_62·a_1_1 + b_2_26·a_5_3 + b_2_28·a_1_1 + b_2_1·b_4_6·c_8_18·b_3_3
       + b_2_12·c_8_18·b_5_6 + b_2_12·c_8_18·a_5_3 + b_2_14·c_8_18·a_1_0
  134. b_9_222 + b_2_2·b_4_64 + b_2_23·b_4_63 + b_2_25·b_4_62 + b_2_29
       + b_2_1·b_4_62·b_8_14 + b_2_13·b_4_6·b_8_14 + b_2_15·b_4_62 + b_4_63·a_1_1·b_5_7
       + b_2_22·b_4_62·a_1_1·b_5_7 + b_2_24·b_4_6·a_1_1·b_5_7 + b_2_27·a_1_1·b_3_2
       + b_2_1·b_4_62·c_8_18 + b_2_13·b_4_6·c_8_18 + b_4_62·c_8_18·a_1_12

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 18.
  • 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_18, a Duflot regular element of degree 8
    2. b_4_6 + b_2_12, an element of degree 4
    3. b_3_4 + b_3_2, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, 4, 9, 12].
  • 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 64

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. b_2_1 → 0, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_3_2 → 0, an element of degree 3
  6. b_3_3 → 0, an element of degree 3
  7. b_3_4 → 0, an element of degree 3
  8. b_4_4 → 0, an element of degree 4
  9. b_4_6 → 0, an element of degree 4
  10. a_5_3 → 0, an element of degree 5
  11. b_5_6 → 0, an element of degree 5
  12. b_5_7 → 0, an element of degree 5
  13. b_6_10 → 0, an element of degree 6
  14. b_6_11 → 0, an element of degree 6
  15. b_7_13 → 0, an element of degree 7
  16. b_7_14 → 0, an element of degree 7
  17. b_8_14 → 0, an element of degree 8
  18. c_8_18 → c_1_08, an element of degree 8
  19. b_9_22 → 0, an element of degree 9

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. b_2_1 → 0, an element of degree 2
  4. b_2_2 → c_1_12, an element of degree 2
  5. b_3_2 → c_1_13, an element of degree 3
  6. b_3_3 → c_1_13, an element of degree 3
  7. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_13, an element of degree 3
  8. b_4_4 → 0, an element of degree 4
  9. b_4_6 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
  10. a_5_3 → 0, an element of degree 5
  11. b_5_6 → 0, an element of degree 5
  12. b_5_7 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_15, an element of degree 5
  13. b_6_10 → c_1_14·c_1_22 + c_1_15·c_1_2, an element of degree 6
  14. b_6_11 → c_1_26 + c_1_12·c_1_24 + c_1_14·c_1_22 + 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
  15. b_7_13 → c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23 + c_1_15·c_1_22
       + c_1_02·c_1_15 + c_1_04·c_1_13, an element of degree 7
  16. b_7_14 → c_1_13·c_1_24 + c_1_15·c_1_22, an element of degree 7
  17. b_8_14 → 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_18, an element of degree 8
  18. c_8_18 → c_1_28 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_17·c_1_2 + 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
  19. b_9_22 → c_1_1·c_1_28 + c_1_13·c_1_26 + c_1_14·c_1_25 + c_1_16·c_1_23
       + c_1_17·c_1_22 + c_1_18·c_1_2, an element of degree 9

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. b_2_1 → c_1_12, an element of degree 2
  4. b_2_2 → 0, an element of degree 2
  5. b_3_2 → 0, an element of degree 3
  6. b_3_3 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  7. b_3_4 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  8. b_4_4 → c_1_12·c_1_22 + c_1_13·c_1_2, an element of degree 4
  9. b_4_6 → c_1_24 + c_1_12·c_1_22, an element of degree 4
  10. a_5_3 → 0, an element of degree 5
  11. b_5_6 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  12. b_5_7 → 0, an element of degree 5
  13. b_6_10 → c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_12·c_1_22
       + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  14. b_6_11 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22
       + c_1_15·c_1_2, an element of degree 6
  15. b_7_13 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23
       + c_1_15·c_1_22 + c_1_16·c_1_2, an element of degree 7
  16. b_7_14 → c_1_13·c_1_24 + c_1_15·c_1_22 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22
       + c_1_04·c_1_12·c_1_2, an element of degree 7
  17. b_8_14 → c_1_16·c_1_22 + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22
       + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2 + c_1_04·c_1_12·c_1_22
       + c_1_04·c_1_13·c_1_2, an element of degree 8
  18. c_8_18 → c_1_28 + c_1_14·c_1_24 + c_1_16·c_1_22 + c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24
       + c_1_0·c_1_15·c_1_22 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2
       + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_08, an element of degree 8
  19. b_9_22 → c_1_1·c_1_28 + c_1_13·c_1_26 + c_1_14·c_1_25 + c_1_15·c_1_24
       + c_1_16·c_1_23 + c_1_17·c_1_22 + c_1_0·c_1_12·c_1_26 + c_1_0·c_1_13·c_1_25
       + c_1_0·c_1_15·c_1_23 + c_1_0·c_1_16·c_1_22 + c_1_02·c_1_1·c_1_26
       + c_1_02·c_1_12·c_1_25 + c_1_02·c_1_13·c_1_24 + c_1_02·c_1_14·c_1_23
       + c_1_02·c_1_15·c_1_22 + c_1_02·c_1_16·c_1_2 + c_1_04·c_1_1·c_1_24
       + c_1_04·c_1_14·c_1_2, an element of degree 9

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 64


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