Cohomology of group number 10 of order 625

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

General information on the group

  • The group has 2 minimal generators and exponent 25.
  • It is non-abelian.
  • It has p-Rank 2.
  • Its center has rank 1.
  • It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t8  −  t7  +  t6  −  t5  +  t4  −  t3  +  t2  +  1
    (t  −  1)2 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

Ring generators

The cohomology ring has 23 minimal generators of maximal degree 15:

  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. a_2_1, a nilpotent element of degree 2
  5. b_2_2, an element of degree 2
  6. a_3_2, a nilpotent element of degree 3
  7. a_4_0, a nilpotent element of degree 4
  8. a_4_1, a nilpotent element of degree 4
  9. a_5_2, a nilpotent element of degree 5
  10. a_6_0, a nilpotent element of degree 6
  11. a_6_2, a nilpotent element of degree 6
  12. a_7_2, a nilpotent element of degree 7
  13. a_8_0, a nilpotent element of degree 8
  14. b_8_2, an element of degree 8
  15. a_9_2, a nilpotent element of degree 9
  16. a_9_3, a nilpotent element of degree 9
  17. b_10_3, an element of degree 10
  18. c_10_4, a Duflot regular element of degree 10
  19. a_11_5, a nilpotent element of degree 11
  20. b_12_6, an element of degree 12
  21. a_13_5, a nilpotent element of degree 13
  22. b_14_6, an element of degree 14
  23. a_15_6, a nilpotent element of degree 15

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

Ring relations

There are 10 "obvious" relations:
   a_1_02, a_1_12, a_3_22, a_5_22, a_7_22, a_9_22, a_9_32, a_11_52, a_13_52, a_15_62

Apart from that, there are 220 minimal relations of maximal degree 29:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_0
  3. a_2_1·a_1_1 − a_2_0·a_1_1
  4. a_2_1·a_1_0 − a_2_0·a_1_1
  5. b_2_2·a_1_1
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9. a_2_0·b_2_2
  10. a_2_1·b_2_2
  11. a_1_1·a_3_2
  12. a_1_0·a_3_2
  13. b_2_2·a_3_2
  14. a_2_0·a_3_2
  15. a_2_1·a_3_2
  16. a_4_0·a_1_1
  17. a_4_0·a_1_0
  18. a_4_1·a_1_0
  19. b_2_2·a_4_0
  20. a_2_0·a_4_0
  21. a_2_1·a_4_0
  22. b_2_2·a_4_1
  23. a_2_0·a_4_1
  24. a_2_1·a_4_1
  25. a_1_1·a_5_2
  26. a_1_0·a_5_2
  27. a_4_0·a_3_2
  28. a_4_1·a_3_2
  29. b_2_2·a_5_2
  30. a_2_0·a_5_2
  31. a_2_1·a_5_2
  32. a_6_0·a_1_1
  33. a_6_0·a_1_0
  34. a_6_2·a_1_0
  35. a_4_02
  36. a_4_0·a_4_1
  37. a_4_12
  38. a_3_2·a_5_2
  39. b_2_2·a_6_0
  40. a_2_0·a_6_0
  41. a_2_1·a_6_0
  42. b_2_2·a_6_2
  43. a_2_0·a_6_2
  44. a_2_1·a_6_2
  45. a_1_1·a_7_2
  46. a_1_0·a_7_2
  47. a_4_0·a_5_2
  48. a_4_1·a_5_2
  49. a_6_0·a_3_2
  50. a_6_2·a_3_2
  51. b_2_2·a_7_2
  52. a_2_0·a_7_2
  53. a_2_1·a_7_2
  54. a_8_0·a_1_1
  55. a_8_0·a_1_0
  56. b_8_2·a_1_0
  57. a_4_0·a_6_0
  58. a_4_1·a_6_0
  59. a_4_0·a_6_2
  60. a_4_1·a_6_2
  61. a_3_2·a_7_2
  62. a_2_0·a_8_0
  63. a_2_1·a_8_0
  64. b_2_2·b_8_2 − 2·b_2_2·a_8_0
  65. a_2_0·b_8_2
  66.  − a_2_1·b_8_2 + a_1_1·a_9_2
  67. a_1_0·a_9_2
  68. a_1_1·a_9_3
  69.  − b_2_2·a_8_0 + a_1_0·a_9_3
  70. a_6_0·a_5_2
  71. a_6_2·a_5_2
  72. a_4_0·a_7_2
  73. a_4_1·a_7_2
  74. a_8_0·a_3_2
  75. b_2_2·a_9_2
  76. a_2_0·a_9_2
  77. a_2_1·a_9_2
  78. a_2_0·a_9_3
  79. a_2_1·a_9_3
  80. b_10_3·a_1_1 − b_8_2·a_3_2
  81. b_10_3·a_1_0
  82. a_6_02
  83. a_6_22
  84. a_6_0·a_6_2
  85. a_5_2·a_7_2
  86. a_4_0·a_8_0
  87. a_4_1·a_8_0
  88. a_4_1·b_8_2 − 2·a_4_0·b_8_2
  89.  − a_4_0·b_8_2 + a_3_2·a_9_2
  90. a_3_2·a_9_3
  91. b_2_2·b_10_3 + 2·b_2_2·a_1_0·a_9_3
  92. a_2_0·b_10_3
  93.  − a_4_0·b_8_2 + a_2_1·b_10_3
  94.  − 2·a_4_0·b_8_2 + a_1_1·a_11_5
  95. a_1_0·a_11_5 − b_2_2·a_1_0·a_9_3
  96. a_6_2·a_7_2
  97. a_6_0·a_7_2
  98. a_8_0·a_5_2
  99. a_4_0·a_9_2
  100. a_4_1·a_9_2
  101. a_4_0·a_9_3
  102. a_4_1·a_9_3
  103. b_10_3·a_3_2 − b_8_2·a_5_2
  104. b_2_2·a_11_5 − b_2_22·a_9_3 + b_2_2·c_10_4·a_1_0
  105. a_2_0·a_11_5 + a_2_0·c_10_4·a_1_1
  106. a_2_1·a_11_5 + 2·a_2_0·c_10_4·a_1_1
  107. b_12_6·a_1_1 − b_8_2·a_5_2 − a_2_0·c_10_4·a_1_1
  108. b_12_6·a_1_0 − b_2_2·c_10_4·a_1_0 + a_2_0·c_10_4·a_1_1
  109. a_6_2·a_8_0
  110. a_6_0·a_8_0
  111. a_6_2·b_8_2
  112. 2·a_6_0·b_8_2 + a_5_2·a_9_2
  113. a_5_2·a_9_3
  114. 2·a_6_0·b_8_2 + a_4_0·b_10_3
  115.  − a_6_0·b_8_2 + a_4_1·b_10_3
  116.  − a_6_0·b_8_2 + a_3_2·a_11_5
  117. b_2_2·b_12_6 + 2·b_2_22·a_1_0·a_9_3 − b_2_22·c_10_4
  118. a_2_0·b_12_6
  119. 2·a_6_0·b_8_2 + a_2_1·b_12_6
  120. 2·a_6_0·b_8_2 + a_1_1·a_13_5
  121. a_1_0·a_13_5
  122. a_8_0·a_7_2
  123. a_6_2·a_9_2
  124. a_6_0·a_9_2
  125. a_6_2·a_9_3
  126. a_6_0·a_9_3
  127. b_10_3·a_5_2 + 2·b_8_2·a_7_2
  128. a_4_0·a_11_5
  129. a_4_1·a_11_5 + a_4_1·c_10_4·a_1_1
  130. b_12_6·a_3_2 + 2·b_8_2·a_7_2
  131. b_2_2·a_13_5 + b_2_22·c_10_4·a_1_0
  132. a_2_0·a_13_5
  133. a_2_1·a_13_5
  134. b_14_6·a_1_1 + 2·b_8_2·a_7_2
  135. b_14_6·a_1_0 + 2·b_2_22·c_10_4·a_1_0
  136. a_8_02
  137.  − a_8_0·b_8_2 + a_7_2·a_9_2
  138. a_7_2·a_9_3
  139. a_6_2·b_10_3
  140.  − a_8_0·b_8_2 + a_6_0·b_10_3
  141.  − a_8_0·b_8_2 + a_5_2·a_11_5
  142. 2·a_8_0·b_8_2 + a_4_0·b_12_6
  143.  − a_8_0·b_8_2 + a_4_1·b_12_6
  144. 2·a_8_0·b_8_2 + a_3_2·a_13_5
  145. b_2_2·b_14_6 + 2·b_2_23·c_10_4
  146. a_2_0·b_14_6
  147. 2·a_8_0·b_8_2 + a_2_1·b_14_6
  148. a_8_0·b_8_2 + a_1_1·a_15_6
  149. a_1_0·a_15_6 − b_2_23·a_1_0·a_9_3
  150. a_8_0·a_9_2
  151. a_8_0·a_9_3
  152. b_8_2·a_9_3 + b_8_22·a_1_1
  153. b_10_3·a_7_2 − b_8_22·a_1_1
  154. a_6_2·a_11_5 + a_6_2·c_10_4·a_1_1
  155. a_6_0·a_11_5
  156. b_12_6·a_5_2 + 2·b_8_22·a_1_1
  157. a_4_0·a_13_5
  158. a_4_1·a_13_5
  159. b_14_6·a_3_2 + 2·b_8_22·a_1_1
  160. b_2_2·a_15_6 − b_2_24·a_9_3 + 2·b_2_23·c_10_4·a_1_0
  161. a_2_0·a_15_6
  162. a_2_1·a_15_6
  163. a_9_2·a_9_3 − b_8_2·a_1_1·a_9_2
  164. a_8_0·b_10_3 − b_8_2·a_1_1·a_9_2
  165. a_7_2·a_11_5 − 2·b_8_2·a_1_1·a_9_2
  166. a_6_2·b_12_6
  167. a_6_0·b_12_6 − b_8_2·a_1_1·a_9_2
  168. a_5_2·a_13_5 + 2·b_8_2·a_1_1·a_9_2
  169. a_4_0·b_14_6 + 2·b_8_2·a_1_1·a_9_2
  170. a_4_1·b_14_6 − b_8_2·a_1_1·a_9_2
  171. a_3_2·a_15_6 + b_8_2·a_1_1·a_9_2
  172. b_10_3·a_9_3 + b_8_22·a_3_2
  173. a_8_0·a_11_5
  174.  − 2·b_10_3·a_9_2 + b_8_2·a_11_5 + 2·b_8_22·a_3_2 + b_8_2·c_10_4·a_1_1
  175. b_12_6·a_7_2 − b_8_22·a_3_2
  176. a_6_2·a_13_5
  177. a_6_0·a_13_5
  178. b_14_6·a_5_2 + 2·b_8_22·a_3_2
  179. a_4_0·a_15_6
  180. a_4_1·a_15_6
  181. a_9_3·a_11_5 + a_9_2·a_11_5 − c_10_4·a_1_1·a_9_2 − c_10_4·a_1_0·a_9_3
  182. a_9_3·a_11_5 + b_8_2·a_1_1·a_11_5 − c_10_4·a_1_0·a_9_3
  183. a_8_0·b_12_6 − 2·a_9_3·a_11_5 + c_10_4·a_1_0·a_9_3
  184.  − b_10_32 + b_8_2·b_12_6 − a_9_3·a_11_5 + c_10_4·a_1_1·a_9_2 − c_10_4·a_1_0·a_9_3
  185.  − 2·a_9_3·a_11_5 + a_7_2·a_13_5 + 2·c_10_4·a_1_0·a_9_3
  186. a_6_2·b_14_6
  187. a_6_0·b_14_6 − 2·a_9_3·a_11_5 + 2·c_10_4·a_1_0·a_9_3
  188. 2·a_9_3·a_11_5 + a_5_2·a_15_6 − 2·c_10_4·a_1_0·a_9_3
  189. b_12_6·a_9_3 + b_8_22·a_5_2 − b_2_2·c_10_4·a_9_3
  190. b_12_6·a_9_2 + 2·b_10_3·a_11_5 − b_8_22·a_5_2 + 2·b_8_2·c_10_4·a_3_2
  191. a_8_0·a_13_5
  192. 2·b_10_3·a_11_5 + b_8_2·a_13_5 − b_8_22·a_5_2 + 2·b_8_2·c_10_4·a_3_2
  193. b_14_6·a_7_2 − b_8_22·a_5_2
  194. a_6_2·a_15_6
  195. a_6_0·a_15_6
  196. a_9_2·a_13_5
  197. a_9_3·a_13_5 + b_8_2·a_1_1·a_13_5 − b_2_2·c_10_4·a_1_0·a_9_3
  198. a_8_0·b_14_6 + a_9_3·a_13_5 + b_2_2·c_10_4·a_1_0·a_9_3
  199.  − b_10_3·b_12_6 + b_8_2·b_14_6 + 2·a_9_3·a_13_5 − c_10_4·a_1_1·a_11_5
  200.  − 2·a_9_3·a_13_5 + a_7_2·a_15_6 + 2·b_2_2·c_10_4·a_1_0·a_9_3
  201. 2·b_12_6·a_11_5 + b_10_3·a_13_5 + 2·b_8_22·a_7_2 + 2·b_8_2·c_10_4·a_5_2
       − 2·b_2_22·c_10_4·a_9_3 + 2·b_2_2·c_10_42·a_1_0
  202. b_14_6·a_9_3 − 2·b_8_22·a_7_2 + 2·b_2_22·c_10_4·a_9_3
  203. b_14_6·a_9_2 + 2·b_12_6·a_11_5 + 2·b_8_22·a_7_2 + 2·b_8_2·c_10_4·a_5_2
       − 2·b_2_22·c_10_4·a_9_3 + 2·b_2_2·c_10_42·a_1_0
  204. a_8_0·a_15_6
  205. b_12_6·a_11_5 + b_8_2·a_15_6 − 2·b_8_2·c_10_4·a_5_2 − b_2_22·c_10_4·a_9_3
       + b_2_2·c_10_42·a_1_0
  206. 2·b_12_62 − b_8_23 + 2·a_11_5·a_13_5 + c_10_4·a_1_1·a_13_5
       + b_2_22·c_10_4·a_1_0·a_9_3 − 2·b_2_22·c_10_42
  207. b_12_62 + b_10_3·b_14_6 − b_8_23 − a_11_5·a_13_5 + b_2_22·c_10_4·a_1_0·a_9_3
       − b_2_22·c_10_42
  208.  − 2·b_12_62 + b_8_23 − a_11_5·a_13_5 + a_9_3·a_15_6 + b_2_22·c_10_4·a_1_0·a_9_3
       + 2·b_2_22·c_10_42
  209. b_12_62 + 2·b_8_23 + a_9_2·a_15_6 − b_2_22·c_10_4·a_1_0·a_9_3 − b_2_22·c_10_42
  210. 2·b_12_62 − b_8_23 + a_11_5·a_13_5 + b_8_2·a_1_1·a_15_6
       + 2·b_2_22·c_10_4·a_1_0·a_9_3 − 2·b_2_22·c_10_42
  211. b_12_6·a_13_5 + 2·b_8_22·a_9_2 + b_2_22·c_10_42·a_1_0
  212. b_14_6·a_11_5 − b_8_22·a_9_2 + b_8_23·a_1_1 − 2·b_8_2·c_10_4·a_7_2
       + 2·b_2_23·c_10_4·a_9_3 − 2·b_2_22·c_10_42·a_1_0
  213. b_10_3·a_15_6 + b_8_22·a_9_2 − b_8_23·a_1_1 + b_8_2·c_10_4·a_7_2
  214. b_12_6·b_14_6 + 2·b_8_22·b_10_3 − 2·b_8_22·a_1_1·a_9_2 + b_2_23·c_10_4·a_1_0·a_9_3
       + 2·b_2_23·c_10_42
  215. 2·a_11_5·a_15_6 + c_10_4·a_1_1·a_15_6 − 2·b_2_23·c_10_4·a_1_0·a_9_3
  216. b_14_6·a_13_5 + b_8_22·a_11_5 + 2·b_8_23·a_3_2 + b_8_22·c_10_4·a_1_1
       − 2·b_2_23·c_10_42·a_1_0
  217. b_12_6·a_15_6 − 2·b_8_22·a_11_5 − b_8_22·c_10_4·a_1_1 − b_2_24·c_10_4·a_9_3
       + 2·b_2_23·c_10_42·a_1_0
  218. b_14_62 + 2·b_8_22·b_12_6 − 2·b_8_22·a_1_1·a_11_5 + b_8_2·c_10_4·a_1_1·a_9_2
       + b_2_24·c_10_42
  219. a_13_5·a_15_6 − 2·b_8_22·a_1_1·a_11_5 − b_8_2·c_10_4·a_1_1·a_9_2
       + b_2_24·c_10_4·a_1_0·a_9_3
  220. b_14_6·a_15_6 + b_8_22·a_13_5 − b_8_23·a_5_2 + b_8_22·c_10_4·a_3_2
       + 2·b_2_25·c_10_4·a_9_3 + b_2_24·c_10_42·a_1_0

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

Data used for Benson′s test

  • Benson′s completion test succeeded in degree 29.
  • 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_10_4, a Duflot regular element of degree 10
    2. b_8_2 + 2·b_2_24, an element of degree 8
  • The Raw Filter Degree Type of that HSOP is [-1, 7, 16].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625

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. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_4_0 → 0, an element of degree 4
  8. a_4_1 → 0, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. a_6_0 → 0, an element of degree 6
  11. a_6_2 → 0, an element of degree 6
  12. a_7_2 → 0, an element of degree 7
  13. a_8_0 → 0, an element of degree 8
  14. b_8_2 → 0, an element of degree 8
  15. a_9_2 → 0, an element of degree 9
  16. a_9_3 → 0, an element of degree 9
  17. b_10_3 → 0, an element of degree 10
  18. c_10_4 →  − c_2_05, an element of degree 10
  19. a_11_5 → 0, an element of degree 11
  20. b_12_6 → 0, an element of degree 12
  21. a_13_5 → 0, an element of degree 13
  22. b_14_6 → 0, an element of degree 14
  23. a_15_6 → 0, an element of degree 15

Restriction map to a maximal el. ab. subgp. of rank 2

  1. a_1_0 → a_1_1, 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. a_2_1 → 0, an element of degree 2
  5. b_2_2 → c_2_2, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_4_0 → 0, an element of degree 4
  8. a_4_1 → 0, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. a_6_0 → 0, an element of degree 6
  11. a_6_2 → 0, an element of degree 6
  12. a_7_2 → 0, an element of degree 7
  13. a_8_0 →  − c_2_23·a_1_0·a_1_1, an element of degree 8
  14. b_8_2 →  − 2·c_2_23·a_1_0·a_1_1, an element of degree 8
  15. a_9_2 → 0, an element of degree 9
  16. a_9_3 → c_2_24·a_1_0 − c_2_1·c_2_23·a_1_1, an element of degree 9
  17. b_10_3 → 2·c_2_24·a_1_0·a_1_1, an element of degree 10
  18. c_10_4 →  − c_2_24·a_1_0·a_1_1 + c_2_1·c_2_24 − c_2_15, an element of degree 10
  19. a_11_5 → c_2_25·a_1_0 − 2·c_2_1·c_2_24·a_1_1 + c_2_15·a_1_1, an element of degree 11
  20. b_12_6 → c_2_25·a_1_0·a_1_1 + c_2_1·c_2_25 − c_2_15·c_2_2, an element of degree 12
  21. a_13_5 →  − c_2_1·c_2_25·a_1_1 + c_2_15·c_2_2·a_1_1, an element of degree 13
  22. b_14_6 → 2·c_2_26·a_1_0·a_1_1 − 2·c_2_1·c_2_26 + 2·c_2_15·c_2_22, an element of degree 14
  23. a_15_6 → c_2_27·a_1_0 + 2·c_2_1·c_2_26·a_1_1 + 2·c_2_15·c_2_22·a_1_1, an element of degree 15

Restriction map to a maximal el. ab. subgp. 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. a_2_1 → 0, an element of degree 2
  5. b_2_2 → 0, an element of degree 2
  6. a_3_2 → 0, an element of degree 3
  7. a_4_0 → 0, an element of degree 4
  8. a_4_1 → 0, an element of degree 4
  9. a_5_2 → 0, an element of degree 5
  10. a_6_0 → 0, an element of degree 6
  11. a_6_2 → 0, an element of degree 6
  12. a_7_2 → 0, an element of degree 7
  13. a_8_0 → 0, an element of degree 8
  14. b_8_2 → 2·c_2_24, an element of degree 8
  15. a_9_2 →  − 2·c_2_24·a_1_1, an element of degree 9
  16. a_9_3 → 0, an element of degree 9
  17. b_10_3 →  − 2·c_2_25, an element of degree 10
  18. c_10_4 →  − c_2_25 + c_2_1·c_2_24 − c_2_15, an element of degree 10
  19. a_11_5 →  − c_2_25·a_1_1, an element of degree 11
  20. b_12_6 → 2·c_2_26, an element of degree 12
  21. a_13_5 →  − 2·c_2_26·a_1_1, an element of degree 13
  22. b_14_6 →  − 2·c_2_27, an element of degree 14
  23. a_15_6 → c_2_27·a_1_1, an element of degree 15

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 625


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