Mod-5-Cohomology of group number 474 of order 2000

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • The group order factors as 24 · 53.
  • It is non-abelian.
  • It has 5-Rank 2.
  • The centre of a Sylow 5-subgroup has rank 1.
  • Its Sylow 5-subgroup has 6 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.

Structure of the cohomology ring

The computation was based on 15 stability conditions for H*(E125; GF(5)).

General information

  • The cohomology ring is of dimension 2 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    1  −  2·t  +  3·t2  −  4·t3  +  5·t4  −  5·t5  +  5·t6  −  3·t7  +  2·t8  −  2·t9  +  2·t10  −  4·t11  +  5·t12  −  4·t13  +  4·t14  −  t15  −  t17  +  3·t18  −  6·t19  +  7·t20  −  6·t21  +  4·t22  −  t23  −  t25  +  3·t26  −  4·t27  +  5·t28  −  4·t29  +  2·t30  −  2·t31  +  2·t32  −  3·t33  +  5·t34  −  5·t35  +  5·t36  −  4·t37  +  3·t38  −  2·t39  +  t40
    ( − 1  +  t)2 · (1  +  t2)2 · (1  −  t  +  t2  −  t3  +  t4) · (1  +  t4) · (1  +  t  +  t2  +  t3  +  t4) · (1  −  t2  +  t4  −  t6  +  t8) · (1  −  t4  +  t8  −  t12  +  t16)
  • The a-invariants are -∞,-16,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 20 minimal generators of maximal degree 40:

  1. a_4_0, a nilpotent element of degree 4
  2. a_5_0, a nilpotent element of degree 5
  3. a_7_1, a nilpotent element of degree 7
  4. a_7_0, a nilpotent element of degree 7
  5. b_8_1, an element of degree 8
  6. b_8_0, an element of degree 8
  7. a_13_1, a nilpotent element of degree 13
  8. b_14_0, an element of degree 14
  9. a_15_2, a nilpotent element of degree 15
  10. a_16_2, a nilpotent element of degree 16
  11. a_18_1, a nilpotent element of degree 18
  12. a_19_1, a nilpotent element of degree 19
  13. a_23_2, a nilpotent element of degree 23
  14. a_24_2, a nilpotent element of degree 24
  15. a_27_3, a nilpotent element of degree 27
  16. b_28_2, an element of degree 28
  17. a_38_1, a nilpotent element of degree 38
  18. a_39_3, a nilpotent element of degree 39
  19. a_39_1, a nilpotent element of degree 39
  20. c_40_2, a Duflot element of degree 40

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 10 "obvious" relations:
   a_5_02, a_7_02, a_7_12, a_13_12, a_15_22, a_19_12, a_23_22, a_27_32, a_39_12, a_39_32

Apart from that, there are 164 minimal relations of maximal degree 78:

  1. a_4_02
  2. a_4_0·a_5_0
  3. a_4_0·a_7_0
  4. a_4_0·a_7_1
  5. a_5_0·a_7_0
  6. a_4_0·b_8_0
  7. a_4_0·b_8_1 + a_5_0·a_7_1
  8. b_8_0·a_5_0
  9. a_7_0·a_7_1
  10. b_8_0·a_7_1 + 2·b_8_0·a_7_0
  11. b_8_1·a_7_0 + b_8_0·a_7_0
  12. b_8_0·b_8_1 + b_8_02
  13. a_4_0·a_13_1
  14. a_4_0·b_14_0 + a_5_0·a_13_1
  15. a_4_0·a_15_2
  16. a_4_0·a_16_2
  17. a_5_0·a_15_2
  18. a_7_0·a_13_1
  19. a_7_1·a_13_1
  20. a_16_2·a_5_0
  21. b_8_0·a_13_1
  22. b_14_0·a_7_0
  23. b_14_0·a_7_1 − b_8_1·a_13_1
  24. a_4_0·a_18_1
  25. a_7_0·a_15_2
  26. a_7_1·a_15_2
  27. b_8_0·b_14_0
  28. a_4_0·a_19_1
  29. a_16_2·a_7_0
  30. a_16_2·a_7_1
  31. a_18_1·a_5_0
  32. b_8_0·a_15_2
  33. b_8_1·a_15_2
  34. a_5_0·a_19_1
  35. b_8_0·a_16_2
  36. b_8_1·a_16_2
  37. a_18_1·a_7_0
  38. a_18_1·a_7_1
  39. a_7_1·a_19_1 + 2·a_7_0·a_19_1
  40. b_8_0·a_18_1 − 2·a_7_0·a_19_1
  41. b_8_1·a_18_1 + 2·a_7_0·a_19_1
  42. a_4_0·a_23_2
  43. b_8_1·a_19_1 + b_8_0·a_19_1
  44. a_4_0·a_24_2
  45. a_5_0·a_23_2
  46. a_13_1·a_15_2
  47. a_16_2·a_13_1
  48. a_24_2·a_5_0
  49. b_14_0·a_15_2
  50. a_7_0·a_23_2
  51. a_7_1·a_23_2
  52. b_14_0·a_16_2
  53. a_4_0·a_27_3
  54. a_16_2·a_15_2
  55. a_18_1·a_13_1
  56. a_24_2·a_7_0
  57. a_24_2·a_7_1
  58. b_8_0·a_23_2
  59. b_8_1·a_23_2
  60. a_16_22
  61. a_5_0·a_27_3
  62. a_13_1·a_19_1
  63. a_4_0·b_28_2
  64. b_8_0·a_24_2
  65. b_8_1·a_24_2
  66. b_14_0·a_18_1
  67. a_18_1·a_15_2
  68. b_14_0·a_19_1
  69. b_28_2·a_5_0
  70. a_16_2·a_18_1
  71. a_7_0·a_27_3
  72. a_7_1·a_27_3
  73. a_15_2·a_19_1
  74. a_16_2·a_19_1
  75. b_8_1·a_27_3 + b_8_0·a_27_3
  76. b_28_2·a_7_0 + b_8_0·a_27_3
  77. b_28_2·a_7_1 − 2·b_8_0·a_27_3
  78. a_18_12
  79. a_13_1·a_23_2
  80. b_8_1·b_28_2 + b_8_0·b_28_2
  81. a_18_1·a_19_1
  82. a_24_2·a_13_1
  83. b_14_0·a_23_2
  84. a_15_2·a_23_2
  85. b_14_0·a_24_2
  86. a_16_2·a_23_2
  87. a_24_2·a_15_2
  88. a_16_2·a_24_2
  89. a_13_1·a_27_3
  90. a_18_1·a_23_2
  91. b_14_0·a_27_3
  92. b_28_2·a_13_1
  93. a_4_0·a_38_1
  94. a_18_1·a_24_2
  95. a_15_2·a_27_3
  96. a_19_1·a_23_2
  97. b_14_0·b_28_2
  98. a_4_0·a_39_1
  99. a_4_0·a_39_3
  100. a_16_2·a_27_3
  101. a_24_2·a_19_1
  102. a_38_1·a_5_0
  103. b_28_2·a_15_2
  104. a_5_0·a_39_1
  105. a_5_0·a_39_3
  106. a_16_2·b_28_2
  107. a_18_1·a_27_3
  108. a_38_1·a_7_0
  109. a_38_1·a_7_1
  110. a_7_0·a_39_3 − a_7_0·a_39_1
  111. a_7_1·a_39_1 + 2·a_7_0·a_39_1 + 2·b_14_02·a_5_0·a_13_1
  112. a_7_1·a_39_3 + 2·a_7_0·a_39_1
  113. a_19_1·a_27_3 + 2·a_7_0·a_39_1
  114. b_8_0·a_38_1 − 2·a_7_0·a_39_1
  115. b_8_1·a_38_1 + 2·a_7_0·a_39_1 + 2·b_14_02·a_5_0·a_13_1
  116. a_18_1·b_28_2 − a_7_0·a_39_1
  117. a_24_2·a_23_2
  118. b_8_0·a_39_3 − b_8_0·a_39_1 + b_8_05·a_7_0
  119. b_8_1·a_39_3 + b_8_0·a_39_1 − b_8_05·a_7_0
  120. b_14_03·a_5_0 + 2·b_8_1·a_39_1 + 2·b_8_0·a_39_1
  121. b_28_2·a_19_1 + 2·b_8_0·a_39_1 − 2·b_8_05·a_7_0
  122. a_24_22
  123. a_23_2·a_27_3
  124. a_24_2·a_27_3
  125. a_38_1·a_13_1
  126. b_28_2·a_23_2
  127. a_13_1·a_39_1 + 2·c_40_2·a_5_0·a_7_1
  128. a_13_1·a_39_3
  129. b_14_0·a_38_1 + 2·c_40_2·a_5_0·a_7_1
  130. a_24_2·b_28_2
  131. a_38_1·a_15_2
  132. b_14_0·a_39_1 − 2·b_8_1·c_40_2·a_5_0
  133. b_14_0·a_39_3
  134. a_16_2·a_38_1
  135. a_15_2·a_39_1
  136. a_15_2·a_39_3
  137. a_16_2·a_39_1
  138. a_16_2·a_39_3
  139. b_14_03·a_13_1 − b_8_1·c_40_2·a_7_1 + 2·b_8_0·c_40_2·a_7_0
  140. b_28_2·a_27_3 + 2·b_8_06·a_7_0 + b_8_0·c_40_2·a_7_0
  141. a_18_1·a_38_1
  142. b_14_04 − 2·b_8_13·b_14_0·a_5_0·a_13_1 − b_8_12·c_40_2 + b_8_02·c_40_2
  143. b_28_22 − 2·b_8_07 − b_8_02·c_40_2
  144. a_18_1·a_39_1
  145. a_18_1·a_39_3
  146. a_38_1·a_19_1
  147. a_19_1·a_39_1 + b_8_04·a_7_0·a_19_1
  148. a_19_1·a_39_3
  149. a_38_1·a_23_2
  150. a_24_2·a_38_1
  151. a_23_2·a_39_1
  152. a_23_2·a_39_3
  153. a_24_2·a_39_1
  154. a_24_2·a_39_3
  155. a_38_1·a_27_3
  156. a_27_3·a_39_1 − b_8_05·a_7_0·a_19_1 + 2·c_40_2·a_7_0·a_19_1
  157. a_27_3·a_39_3 − b_8_05·a_7_0·a_19_1 + 2·c_40_2·a_7_0·a_19_1
  158. b_28_2·a_38_1 + 2·b_8_05·a_7_0·a_19_1 + c_40_2·a_7_0·a_19_1
  159. b_28_2·a_39_1 + b_8_05·a_27_3 + b_8_06·a_19_1 − 2·b_8_0·c_40_2·a_19_1
  160. b_28_2·a_39_3 + b_8_06·a_19_1 − 2·b_8_0·c_40_2·a_19_1
  161. a_38_12
  162. a_38_1·a_39_1
  163. a_38_1·a_39_3
  164. a_39_1·a_39_3 − b_8_04·a_7_0·a_39_1

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 78 using the Hilbert-Poincaré criterion.
  • 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_40_2, an element of degree 40
    2. b_8_1, an element of degree 8
  • A Duflot regular sequence is given by c_40_2.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 24, 46].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(E125; GF(5))

  1. a_4_0 → a_1_1·a_3_5 + 2·a_1_1·a_3_4 + 2·a_1_0·a_3_5
  2. a_5_0 → b_2_3·a_3_5 − b_2_2·a_3_5 + 2·b_2_2·b_2_3·a_1_1 − 2·b_2_22·a_1_1
  3. a_7_1 → b_2_33·a_1_1 − 2·b_2_2·b_2_32·a_1_1 − 2·b_2_22·b_2_3·a_1_1 − 2·b_2_23·a_1_1
       − 2·b_2_23·a_1_0
  4. a_7_0 → a_7_8 + 2·b_2_2·b_2_3·a_3_5 − 2·b_2_22·a_3_5 + 2·b_2_22·a_3_4 + b_2_22·b_2_3·a_1_1
       − 2·b_2_23·a_1_1 + b_2_23·a_1_0
  5. b_8_1 → b_2_34 − 2·b_2_2·b_2_33 − 2·b_2_22·b_2_32 − 2·b_2_23·b_2_3 − 2·b_2_24
  6. b_8_0 → b_8_9 − b_2_2·b_2_33 + b_2_22·b_2_32 − 2·b_2_23·b_2_3 + 2·b_2_24
       + b_2_2·b_2_3·a_1_1·a_3_5 + b_2_22·a_1_1·a_3_5 + 2·b_2_22·a_1_0·a_3_4
  7. a_13_1 → b_2_23·b_2_33·a_1_1 − 2·b_2_24·b_2_32·a_1_1 + 2·b_2_25·b_2_3·a_1_1
       − b_2_26·a_1_1 + 2·b_2_3·c_10_12·a_1_1 − 2·b_2_2·c_10_12·a_1_1
  8. b_14_0 → b_2_23·b_2_34 − 2·b_2_24·b_2_33 + 2·b_2_25·b_2_32 − b_2_26·b_2_3
       − b_2_35·a_1_1·a_3_5 − b_2_24·b_2_3·a_1_1·a_3_5 + b_2_25·a_1_1·a_3_5
       + b_2_25·a_1_0·a_3_5 + 2·b_2_32·c_10_12 − 2·b_2_2·b_2_3·c_10_12
  9. a_15_2 → a_2_0·c_10_12·a_3_5
  10. a_16_2 → c_10_12·a_3_4·a_3_5 − 2·b_2_2·c_10_12·a_1_1·a_3_5 − 2·b_2_2·c_10_12·a_1_0·a_3_5
  11. a_18_1 → b_2_2·b_2_3·c_10_12·a_1_1·a_3_5 + b_2_22·c_10_12·a_1_1·a_3_5
       + b_2_22·c_10_12·a_1_0·a_3_5 + b_2_22·c_10_12·a_1_0·a_3_4
  12. a_19_1 → b_2_2·b_2_32·c_10_12·a_3_5 + b_2_22·b_2_3·c_10_12·a_3_5
       + 2·b_2_22·b_2_32·c_10_12·a_1_1 + b_2_23·c_10_12·a_3_5 + b_2_23·c_10_12·a_3_4
       + 2·b_2_23·b_2_3·c_10_12·a_1_1
  13. a_23_2 → a_2_0·c_10_122·a_1_1
  14. a_24_2 → c_10_122·a_1_1·a_3_4 − c_10_122·a_1_0·a_3_5
  15. a_27_3 → b_2_2·b_2_32·c_10_122·a_1_1 + b_2_22·b_2_3·c_10_122·a_1_1
       + b_2_23·c_10_122·a_1_1 + b_2_23·c_10_122·a_1_0
  16. b_28_2 → b_2_25·b_2_32·c_10_12·a_1_1·a_3_5 − 2·b_2_26·b_2_3·c_10_12·a_1_1·a_3_5
       − 2·b_2_27·c_10_12·a_1_1·a_3_5 − 2·b_2_27·c_10_12·a_1_0·a_3_5
       + 2·b_2_27·c_10_12·a_1_0·a_3_4 − 2·b_2_2·b_2_33·c_10_122
       − 2·b_2_22·b_2_32·c_10_122 − 2·b_2_23·b_2_3·c_10_122 − 2·b_2_24·c_10_122
  17. a_38_1 → b_2_215·b_2_32·a_1_1·a_3_5 − 2·b_2_216·b_2_3·a_1_1·a_3_5 + b_2_217·a_1_1·a_3_5
       + b_2_210·b_2_32·c_10_12·a_1_1·a_3_5 − 2·b_2_211·b_2_3·c_10_12·a_1_1·a_3_5
       − 2·b_2_212·c_10_12·a_1_1·a_3_5 − 2·b_2_212·c_10_12·a_1_0·a_3_5
       − 2·b_2_25·b_2_32·c_10_122·a_1_1·a_3_5 − b_2_26·b_2_3·c_10_122·a_1_1·a_3_5
       − 2·b_2_27·c_10_122·a_1_1·a_3_5 + b_2_32·c_10_123·a_1_1·a_3_5
       − 2·b_2_2·b_2_3·c_10_123·a_1_1·a_3_5 − 2·b_2_22·c_10_123·a_1_1·a_3_5
       − 2·b_2_22·c_10_123·a_1_0·a_3_5 − 2·b_2_22·c_10_123·a_1_0·a_3_4
  18. a_39_3 → c_10_123·a_9_11 − b_2_33·c_10_123·a_3_5 − b_2_2·b_2_33·c_10_123·a_1_1
       + 2·b_2_22·b_2_3·c_10_123·a_3_5 + 2·b_2_22·b_2_32·c_10_123·a_1_1
       + 2·b_2_23·c_10_123·a_3_5 − 2·b_2_23·b_2_3·c_10_123·a_1_1
       + 2·b_2_24·c_10_123·a_1_1
  19. a_39_1 → b_2_215·b_2_33·a_3_5 − 2·b_2_216·b_2_32·a_3_5 + b_2_217·b_2_3·a_3_5
       + b_2_218·b_2_3·a_1_1 − b_2_219·a_1_1 + b_2_219·a_1_0
       + b_2_210·b_2_33·c_10_12·a_3_5 − 2·b_2_211·b_2_32·c_10_12·a_3_5
       + 2·b_2_211·b_2_33·c_10_12·a_1_1 − 2·b_2_212·b_2_3·c_10_12·a_3_5
       + b_2_212·b_2_32·c_10_12·a_1_1 − 2·b_2_213·c_10_12·a_3_5
       + b_2_213·b_2_3·c_10_12·a_1_1 + b_2_214·c_10_12·a_1_1
       − 2·b_2_25·b_2_33·c_10_122·a_3_5 − b_2_26·b_2_32·c_10_122·a_3_5
       + b_2_26·b_2_33·c_10_122·a_1_1 − 2·b_2_27·b_2_3·c_10_122·a_3_5
       − 2·b_2_27·b_2_32·c_10_122·a_1_1 + b_2_28·b_2_3·c_10_122·a_1_1
       + b_2_33·c_10_123·a_3_5 − 2·b_2_2·b_2_32·c_10_123·a_3_5
       + 2·b_2_2·b_2_33·c_10_123·a_1_1 − 2·b_2_22·b_2_3·c_10_123·a_3_5
       + b_2_22·b_2_32·c_10_123·a_1_1 − 2·b_2_23·c_10_123·a_3_5
       − 2·b_2_23·c_10_123·a_3_4 + b_2_23·b_2_3·c_10_123·a_1_1
  20. c_40_2 → b_2_220 + 2·b_2_217·b_2_3·a_1_1·a_3_5 + b_2_218·a_1_1·a_3_5
       + 2·b_2_218·a_1_0·a_3_5 − b_2_211·b_2_34·c_10_12 + 2·b_2_212·b_2_33·c_10_12
       − b_2_213·b_2_32·c_10_12 + b_2_211·b_2_32·c_10_12·a_1_1·a_3_5
       + 2·b_2_212·b_2_3·c_10_12·a_1_1·a_3_5 − 2·b_2_213·c_10_12·a_1_1·a_3_5
       − b_2_213·c_10_12·a_1_0·a_3_5 + 2·b_2_26·b_2_34·c_10_122
       + b_2_27·b_2_33·c_10_122 + b_2_28·b_2_32·c_10_122 + b_2_29·b_2_3·c_10_122
       − 2·b_2_38·c_10_122·a_1_1·a_3_5 + 2·b_2_26·b_2_32·c_10_122·a_1_1·a_3_5
       + 2·b_2_27·b_2_3·c_10_122·a_1_1·a_3_5 − b_2_28·c_10_122·a_1_1·a_3_5
       − b_2_28·c_10_122·a_1_0·a_3_5 − b_2_2·b_2_34·c_10_123
       + 2·b_2_22·b_2_33·c_10_123 − b_2_23·b_2_32·c_10_123
       − 2·b_2_33·c_10_123·a_1_1·a_3_5 + 2·b_2_2·b_2_32·c_10_123·a_1_1·a_3_5
       + 2·b_2_22·b_2_3·c_10_123·a_1_1·a_3_5 + 2·b_2_23·c_10_123·a_1_1·a_3_5
       − 2·b_2_23·c_10_123·a_1_0·a_3_4 + c_10_124

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

  1. a_4_0 → 0, an element of degree 4
  2. a_5_0 → 0, an element of degree 5
  3. a_7_1 → 0, an element of degree 7
  4. a_7_0 → 0, an element of degree 7
  5. b_8_1 → 0, an element of degree 8
  6. b_8_0 → 0, an element of degree 8
  7. a_13_1 → 0, an element of degree 13
  8. b_14_0 → 0, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 → 0, an element of degree 18
  12. a_19_1 → 0, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 → 0, an element of degree 27
  16. b_28_2 → 0, an element of degree 28
  17. a_38_1 → 0, an element of degree 38
  18. a_39_3 → 0, an element of degree 39
  19. a_39_1 → 0, an element of degree 39
  20. c_40_2 → c_2_020, an element of degree 40

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

  1. a_4_0 → 0, an element of degree 4
  2. a_5_0 → 0, an element of degree 5
  3. a_7_1 →  − 2·c_2_23·a_1_1, an element of degree 7
  4. a_7_0 → c_2_23·a_1_1, an element of degree 7
  5. b_8_1 →  − 2·c_2_24, an element of degree 8
  6. b_8_0 → 2·c_2_24, an element of degree 8
  7. a_13_1 → 0, an element of degree 13
  8. b_14_0 → 0, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 → c_2_1·c_2_27·a_1_0·a_1_1 − c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
  12. a_19_1 →  − c_2_1·c_2_28·a_1_0 + c_2_12·c_2_27·a_1_1 + c_2_15·c_2_24·a_1_0
       − c_2_16·c_2_23·a_1_1, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 → c_2_12·c_2_211·a_1_1 − 2·c_2_16·c_2_27·a_1_1 + c_2_110·c_2_23·a_1_1, an element of degree 27
  16. b_28_2 →  − 2·c_2_12·c_2_212 − c_2_16·c_2_28 − 2·c_2_110·c_2_24, an element of degree 28
  17. a_38_1 →  − 2·c_2_13·c_2_215·a_1_0·a_1_1 + c_2_17·c_2_211·a_1_0·a_1_1
       − c_2_111·c_2_27·a_1_0·a_1_1 + 2·c_2_115·c_2_23·a_1_0·a_1_1, an element of degree 38
  18. a_39_3 → 2·c_2_13·c_2_216·a_1_0 − 2·c_2_14·c_2_215·a_1_1 − c_2_17·c_2_212·a_1_0
       + c_2_18·c_2_211·a_1_1 + c_2_111·c_2_28·a_1_0 − c_2_112·c_2_27·a_1_1
       − 2·c_2_115·c_2_24·a_1_0 + 2·c_2_116·c_2_23·a_1_1, an element of degree 39
  19. a_39_1 → c_2_219·a_1_1 + 2·c_2_13·c_2_216·a_1_0 − 2·c_2_14·c_2_215·a_1_1
       − c_2_17·c_2_212·a_1_0 + c_2_18·c_2_211·a_1_1 + c_2_111·c_2_28·a_1_0
       − c_2_112·c_2_27·a_1_1 − 2·c_2_115·c_2_24·a_1_0 + 2·c_2_116·c_2_23·a_1_1, an element of degree 39
  20. c_40_2 → c_2_220 + c_2_14·c_2_216 + c_2_18·c_2_212 + c_2_112·c_2_28
       + c_2_116·c_2_24 + c_2_120, an element of degree 40

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

  1. a_4_0 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
  2. a_5_0 → c_2_22·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
  3. a_7_1 → c_2_23·a_1_1, an element of degree 7
  4. a_7_0 → 0, an element of degree 7
  5. b_8_1 → c_2_24, an element of degree 8
  6. b_8_0 → 0, an element of degree 8
  7. a_13_1 →  − 2·c_2_1·c_2_25·a_1_1 + 2·c_2_15·c_2_2·a_1_1, an element of degree 13
  8. b_14_0 →  − 2·c_2_1·c_2_26 + 2·c_2_15·c_2_22, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 → 0, an element of degree 18
  12. a_19_1 → 0, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 → 0, an element of degree 27
  16. b_28_2 → 0, an element of degree 28
  17. a_38_1 → c_2_13·c_2_215·a_1_0·a_1_1 + 2·c_2_17·c_2_211·a_1_0·a_1_1
       − 2·c_2_111·c_2_27·a_1_0·a_1_1 − c_2_115·c_2_23·a_1_0·a_1_1, an element of degree 38
  18. a_39_3 → 0, an element of degree 39
  19. a_39_1 →  − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
       + 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
       + c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
  20. c_40_2 → 2·c_2_12·c_2_217·a_1_0·a_1_1 + c_2_16·c_2_213·a_1_0·a_1_1
       + 2·c_2_110·c_2_29·a_1_0·a_1_1 + c_2_14·c_2_216 + c_2_18·c_2_212
       + c_2_112·c_2_28 + c_2_116·c_2_24 + c_2_120, an element of degree 40

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

  1. a_4_0 → 0, an element of degree 4
  2. a_5_0 → 0, an element of degree 5
  3. a_7_1 →  − 2·c_2_23·a_1_1, an element of degree 7
  4. a_7_0 → c_2_23·a_1_1, an element of degree 7
  5. b_8_1 →  − 2·c_2_24, an element of degree 8
  6. b_8_0 → 2·c_2_24, an element of degree 8
  7. a_13_1 → 0, an element of degree 13
  8. b_14_0 → 0, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 →  − c_2_1·c_2_27·a_1_0·a_1_1 + c_2_15·c_2_23·a_1_0·a_1_1, an element of degree 18
  12. a_19_1 → c_2_1·c_2_28·a_1_0 − c_2_12·c_2_27·a_1_1 − c_2_15·c_2_24·a_1_0
       + c_2_16·c_2_23·a_1_1, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 →  − c_2_12·c_2_211·a_1_1 + 2·c_2_16·c_2_27·a_1_1 − c_2_110·c_2_23·a_1_1, an element of degree 27
  16. b_28_2 → 2·c_2_12·c_2_212 + c_2_16·c_2_28 + 2·c_2_110·c_2_24, an element of degree 28
  17. a_38_1 →  − 2·c_2_13·c_2_215·a_1_0·a_1_1 + c_2_17·c_2_211·a_1_0·a_1_1
       − c_2_111·c_2_27·a_1_0·a_1_1 + 2·c_2_115·c_2_23·a_1_0·a_1_1, an element of degree 38
  18. a_39_3 → 2·c_2_13·c_2_216·a_1_0 − 2·c_2_14·c_2_215·a_1_1 − c_2_17·c_2_212·a_1_0
       + c_2_18·c_2_211·a_1_1 + c_2_111·c_2_28·a_1_0 − c_2_112·c_2_27·a_1_1
       − 2·c_2_115·c_2_24·a_1_0 + 2·c_2_116·c_2_23·a_1_1, an element of degree 39
  19. a_39_1 → c_2_219·a_1_1 + 2·c_2_13·c_2_216·a_1_0 − 2·c_2_14·c_2_215·a_1_1
       − c_2_17·c_2_212·a_1_0 + c_2_18·c_2_211·a_1_1 + c_2_111·c_2_28·a_1_0
       − c_2_112·c_2_27·a_1_1 − 2·c_2_115·c_2_24·a_1_0 + 2·c_2_116·c_2_23·a_1_1, an element of degree 39
  20. c_40_2 → c_2_220 + c_2_14·c_2_216 + c_2_18·c_2_212 + c_2_112·c_2_28
       + c_2_116·c_2_24 + c_2_120, an element of degree 40

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

  1. a_4_0 → c_2_2·a_1_0·a_1_1, an element of degree 4
  2. a_5_0 →  − c_2_22·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
  3. a_7_1 → c_2_23·a_1_1, an element of degree 7
  4. a_7_0 → 0, an element of degree 7
  5. b_8_1 → c_2_24, an element of degree 8
  6. b_8_0 → 0, an element of degree 8
  7. a_13_1 → 2·c_2_1·c_2_25·a_1_1 − 2·c_2_15·c_2_2·a_1_1, an element of degree 13
  8. b_14_0 → 2·c_2_1·c_2_26 − 2·c_2_15·c_2_22, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 → 0, an element of degree 18
  12. a_19_1 → 0, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 → 0, an element of degree 27
  16. b_28_2 → 0, an element of degree 28
  17. a_38_1 → c_2_13·c_2_215·a_1_0·a_1_1 + 2·c_2_17·c_2_211·a_1_0·a_1_1
       − 2·c_2_111·c_2_27·a_1_0·a_1_1 − c_2_115·c_2_23·a_1_0·a_1_1, an element of degree 38
  18. a_39_3 → 0, an element of degree 39
  19. a_39_1 →  − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
       + 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
       + c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
  20. c_40_2 →  − 2·c_2_12·c_2_217·a_1_0·a_1_1 − c_2_16·c_2_213·a_1_0·a_1_1
       − 2·c_2_110·c_2_29·a_1_0·a_1_1 + c_2_14·c_2_216 + c_2_18·c_2_212
       + c_2_112·c_2_28 + c_2_116·c_2_24 + c_2_120, an element of degree 40

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

  1. a_4_0 →  − 2·c_2_2·a_1_0·a_1_1, an element of degree 4
  2. a_5_0 → 2·c_2_22·a_1_0 − 2·c_2_1·c_2_2·a_1_1, an element of degree 5
  3. a_7_1 → c_2_23·a_1_1, an element of degree 7
  4. a_7_0 → 0, an element of degree 7
  5. b_8_1 → c_2_24, an element of degree 8
  6. b_8_0 → 0, an element of degree 8
  7. a_13_1 → c_2_1·c_2_25·a_1_1 − c_2_15·c_2_2·a_1_1, an element of degree 13
  8. b_14_0 → c_2_1·c_2_26 − c_2_15·c_2_22, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 → 0, an element of degree 18
  12. a_19_1 → 0, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 → 0, an element of degree 27
  16. b_28_2 → 0, an element of degree 28
  17. a_38_1 → c_2_13·c_2_215·a_1_0·a_1_1 + 2·c_2_17·c_2_211·a_1_0·a_1_1
       − 2·c_2_111·c_2_27·a_1_0·a_1_1 − c_2_115·c_2_23·a_1_0·a_1_1, an element of degree 38
  18. a_39_3 → 0, an element of degree 39
  19. a_39_1 →  − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
       + 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
       + c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
  20. c_40_2 → c_2_12·c_2_217·a_1_0·a_1_1 − 2·c_2_16·c_2_213·a_1_0·a_1_1
       + c_2_110·c_2_29·a_1_0·a_1_1 + c_2_14·c_2_216 + c_2_18·c_2_212
       + c_2_112·c_2_28 + c_2_116·c_2_24 + c_2_120, an element of degree 40

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

  1. a_4_0 → 2·c_2_2·a_1_0·a_1_1, an element of degree 4
  2. a_5_0 →  − 2·c_2_22·a_1_0 + 2·c_2_1·c_2_2·a_1_1, an element of degree 5
  3. a_7_1 → c_2_23·a_1_1, an element of degree 7
  4. a_7_0 → 0, an element of degree 7
  5. b_8_1 → c_2_24, an element of degree 8
  6. b_8_0 → 0, an element of degree 8
  7. a_13_1 →  − c_2_1·c_2_25·a_1_1 + c_2_15·c_2_2·a_1_1, an element of degree 13
  8. b_14_0 →  − c_2_1·c_2_26 + c_2_15·c_2_22, an element of degree 14
  9. a_15_2 → 0, an element of degree 15
  10. a_16_2 → 0, an element of degree 16
  11. a_18_1 → 0, an element of degree 18
  12. a_19_1 → 0, an element of degree 19
  13. a_23_2 → 0, an element of degree 23
  14. a_24_2 → 0, an element of degree 24
  15. a_27_3 → 0, an element of degree 27
  16. b_28_2 → 0, an element of degree 28
  17. a_38_1 → c_2_13·c_2_215·a_1_0·a_1_1 + 2·c_2_17·c_2_211·a_1_0·a_1_1
       − 2·c_2_111·c_2_27·a_1_0·a_1_1 − c_2_115·c_2_23·a_1_0·a_1_1, an element of degree 38
  18. a_39_3 → 0, an element of degree 39
  19. a_39_1 →  − c_2_13·c_2_216·a_1_0 + c_2_14·c_2_215·a_1_1 − 2·c_2_17·c_2_212·a_1_0
       + 2·c_2_18·c_2_211·a_1_1 + 2·c_2_111·c_2_28·a_1_0 − 2·c_2_112·c_2_27·a_1_1
       + c_2_115·c_2_24·a_1_0 − c_2_116·c_2_23·a_1_1, an element of degree 39
  20. c_40_2 →  − c_2_12·c_2_217·a_1_0·a_1_1 + 2·c_2_16·c_2_213·a_1_0·a_1_1
       − c_2_110·c_2_29·a_1_0·a_1_1 + c_2_14·c_2_216 + c_2_18·c_2_212
       + c_2_112·c_2_28 + c_2_116·c_2_24 + c_2_120, an element of degree 40

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010