David J. Green

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Cohomology of group number 6 of order 243

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243

General information on the group

• The group has 2 minimal generators and exponent 9.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• 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 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t2  +  1) · (t5  +  2·t3  +  1)

  (t  +  1) · (t  −  1)3 · (t2  −  t  +  1)2 · (t2  +  t  +  1)2

• 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 243

Ring generators

The cohomology ring has 27 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_1, a nilpotent element of degree 2
4. b_2_0, an element of degree 2
5. b_2_2, an element of degree 2
6. a_3_1, a nilpotent element of degree 3
7. a_3_2, a nilpotent element of degree 3
8. a_3_3, a nilpotent element of degree 3
9. a_3_4, a nilpotent element of degree 3
10. a_3_5, a nilpotent element of degree 3
11. a_4_4, a nilpotent element of degree 4
12. a_4_6, a nilpotent element of degree 4
13. b_4_7, an element of degree 4
14. a_5_6, a nilpotent element of degree 5
15. a_5_7, a nilpotent element of degree 5
16. a_5_8, a nilpotent element of degree 5
17. a_5_9, a nilpotent element of degree 5
18. a_5_10, a nilpotent element of degree 5
19. a_6_7, a nilpotent element of degree 6
20. a_6_10, a nilpotent element of degree 6
21. a_6_9, a nilpotent element of degree 6
22. b_6_12, an element of degree 6
23. c_6_14, a Duflot regular element of degree 6
24. c_6_15, a Duflot regular element of degree 6
25. a_7_17, a nilpotent element of degree 7
26. a_7_18, a nilpotent element of degree 7
27. a_8_15, a nilpotent element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243

Ring relations

There are 14 "obvious" relations:
   a_1_0², a_1_1², a_3_1², a_3_2², a_3_3², a_3_4², a_3_5², a_5_6², a_5_7², a_5_8², a_5_9², a_5_10², a_7_17², a_7_18²

Apart from that, there are 276 minimal relations of maximal degree 16:

1. a_1_0·a_1_1
2. a_2_1·a_1_1
3. a_2_1·a_1_0
4. b_2_0·a_1_1
5. b_2_2·a_1_1
6. b_2_2·a_1_0
7. a_2_1²
8. a_2_1·b_2_0
9.  − a_2_1·b_2_2 + a_1_1·a_3_1
10.  − b_2_0·b_2_2 + a_1_0·a_3_1
11. a_1_1·a_3_2
12.  − a_2_1·b_2_2 + a_1_0·a_3_2
13. a_1_1·a_3_3
14. a_1_0·a_3_3
15.  − a_2_1·b_2_2 + a_1_1·a_3_4
16. a_1_0·a_3_4
17. a_1_1·a_3_5
18. b_2_2·a_3_1
19. a_2_1·a_3_1
20. b_2_0·a_3_2
21. a_2_1·a_3_2
22. b_2_2·a_3_3 − b_2_2·a_3_2
23. a_2_1·a_3_3
24. b_2_0·a_3_4
25. b_2_2·a_3_4
26. a_2_1·a_3_4
27. a_2_1·a_3_5
28. a_4_4·a_1_1
29. a_4_4·a_1_0
30. a_4_6·a_1_1
31. a_4_6·a_1_0
32. b_4_7·a_1_1
33. b_4_7·a_1_0 − b_2_0·a_3_3
34. a_3_1·a_3_2
35.  − a_3_2·a_3_3 + a_3_1·a_3_4
36. a_3_3·a_3_4 + a_3_2·a_3_4
37. a_3_4·a_3_5 − a_3_3·a_3_4
38. a_3_2·a_3_5 + a_3_2·a_3_3
39. b_2_0·a_4_4 − b_2_0·a_1_0·a_3_5
40. b_2_2·a_4_4 + a_3_3·a_3_4
41. a_2_1·a_4_4
42. b_2_0·a_4_6 − b_2_0·a_1_0·a_3_5
43. b_2_2·a_4_6 + a_3_2·a_3_3
44. a_2_1·a_4_6
45. b_2_2·b_4_7 − b_2_2³ − a_3_3·a_3_4 + a_3_1·a_3_3
46. a_2_1·b_4_7
47. a_1_1·a_5_6
48.  − a_3_3·a_3_5 + a_1_0·a_5_6 + b_2_0·a_1_0·a_3_5
49. a_3_2·a_3_3 + a_1_1·a_5_7
50. a_3_1·a_3_3 + a_1_0·a_5_7 + b_2_0·a_1_0·a_3_5
51. a_3_3·a_3_4 − a_3_2·a_3_3 + a_1_1·a_5_8
52. a_3_3·a_3_5 + a_3_2·a_3_3 − a_3_1·a_3_3 + a_1_0·a_5_8 + b_2_0·a_1_0·a_3_5
53. a_3_3·a_3_4 + a_1_1·a_5_9
54.  − a_3_3·a_3_5 − a_3_2·a_3_3 + a_1_0·a_5_9 − b_2_0·a_1_0·a_3_5
55. a_3_3·a_3_4 + a_1_1·a_5_10
56.  − a_3_3·a_3_5 − a_3_3·a_3_4 + a_3_2·a_3_3 − a_3_1·a_3_3 + a_1_0·a_5_10
57. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_4_4·a_3_3
58. a_4_4·a_3_1 + a_1_0·a_3_1·a_3_5
59.  − b_2_2²·a_3_5 − b_2_2²·a_3_2 + a_4_4·a_3_5
60. a_4_4·a_3_4
61.  − b_2_2²·a_3_5 − b_2_2²·a_3_2 + a_4_4·a_3_2
62. a_4_6·a_3_3
63. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_4_6·a_3_1 + a_1_0·a_3_1·a_3_5
64. a_4_6·a_3_5
65. a_4_6·a_3_4
66. a_4_6·a_3_2
67. b_4_7·a_3_4
68. b_4_7·a_3_2 − b_2_2²·a_3_5 + b_2_2²·a_3_2
69.  − b_4_7·a_3_5 − b_4_7·a_3_3 + b_2_2²·a_3_5 + b_2_2²·a_3_2 + b_2_0·a_5_6 + b_2_0²·a_3_5
       + b_2_0²·a_3_3 − a_1_0·a_3_1·a_3_5
70. a_2_1·a_5_6
71.  − b_4_7·a_3_1 + b_2_0·a_5_7 + b_2_0²·a_3_5 + b_2_0²·a_3_3 + a_1_0·a_3_1·a_3_5
72. b_2_2·a_5_7 − b_2_2²·a_3_5 + a_1_0·a_3_1·a_3_5
73. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_2_1·a_5_7
74. b_4_7·a_3_5 + b_4_7·a_3_3 + b_4_7·a_3_1 + b_2_0·a_5_8 + b_2_0²·a_3_5 + a_1_0·a_3_1·a_3_5
75.  − b_2_2²·a_3_5 − b_2_2²·a_3_2 + a_2_1·a_5_8
76.  − b_4_7·a_3_5 − b_2_2²·a_3_5 + b_2_2²·a_3_2 + b_2_0·a_5_9 − b_2_0²·a_3_5 − b_2_0²·a_3_3
77. b_2_2·a_5_9 − b_2_2·a_5_6 + b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_1_0·a_3_1·a_3_5
78. a_2_1·a_5_9
79.  − b_4_7·a_3_5 − b_4_7·a_3_3 + b_4_7·a_3_1 − b_2_2²·a_3_5 − b_2_2²·a_3_2 + b_2_0·a_5_10
       + b_2_0²·a_3_3 − a_1_0·a_3_1·a_3_5
80. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_2_1·a_5_10
81. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_6_7·a_1_1
82. a_6_7·a_1_0 + a_1_0·a_3_1·a_3_5
83. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_6_10·a_1_1
84. a_6_10·a_1_0 − a_1_0·a_3_1·a_3_5
85. a_6_9·a_1_1
86. b_2_2²·a_3_5 + b_2_2²·a_3_2 + a_6_9·a_1_0 + a_1_0·a_3_1·a_3_5
87. b_6_12·a_1_1
88. b_6_12·a_1_0 + b_4_7·a_3_3 + b_2_2²·a_3_5 + a_1_0·a_3_1·a_3_5
89. a_4_4²
90. a_4_4·a_4_6
91. a_4_6²
92. a_4_6·b_4_7 − a_4_4·b_4_7
93.  − a_4_4·b_4_7 + b_2_0·a_1_0·a_5_6 + b_2_0²·a_1_0·a_3_5
94. a_3_5·a_5_6 + a_3_3·a_5_6
95. a_3_4·a_5_6
96. a_3_2·a_5_6
97. a_3_1·a_5_7 + b_2_0·a_3_1·a_3_5 − b_2_0·a_1_0·a_5_7 − b_2_0²·a_1_0·a_3_5
98. a_3_5·a_5_7 + a_3_3·a_5_7 + a_3_1·a_5_6 + b_2_0·a_3_1·a_3_5 − b_2_0·a_1_0·a_5_7
       − b_2_0²·a_1_0·a_3_5
99. a_3_2·a_5_7
100. a_3_1·a_5_8 + a_3_1·a_5_6 − b_2_0·a_3_1·a_3_5 − b_2_0·a_1_0·a_5_7 − b_2_0²·a_1_0·a_3_5
101. a_4_4·b_4_7 + a_3_5·a_5_8 + a_3_3·a_5_8 − a_3_1·a_5_6 − b_2_0·a_3_1·a_3_5
        + b_2_0·a_1_0·a_5_7 + b_2_0²·a_1_0·a_3_5
102. a_3_4·a_5_8 + a_3_4·a_5_7
103. a_3_4·a_5_7 − a_3_3·a_5_8 − a_3_3·a_5_7 − a_3_3·a_5_6 + a_3_2·a_5_8
104. a_4_4·b_4_7 + a_3_4·a_5_7 + a_3_3·a_5_9 − a_3_3·a_5_6
105.  − a_4_4·b_4_7 − a_3_3·a_5_7 + a_3_1·a_5_9 − a_3_1·a_5_6 + b_2_0·a_3_1·a_3_5
        − b_2_0·a_1_0·a_5_7 − b_2_0²·a_1_0·a_3_5
106. a_4_4·b_4_7 + a_3_5·a_5_9 − a_3_4·a_5_7
107. a_3_4·a_5_9
108.  − a_3_4·a_5_7 + a_3_2·a_5_9
109.  − a_3_4·a_5_7 + a_3_1·a_5_10 − a_3_1·a_5_6 − b_2_0·a_3_1·a_3_5
110.  − a_4_4·b_4_7 + a_3_5·a_5_10 + a_3_3·a_5_10 − a_3_1·a_5_6 − b_2_0·a_3_1·a_3_5
        + b_2_0·a_1_0·a_5_7 + b_2_0²·a_1_0·a_3_5
111. a_3_4·a_5_10 − a_3_4·a_5_7
112. a_3_4·a_5_7 − a_3_3·a_5_10 − a_3_3·a_5_7 + a_3_3·a_5_6 + a_3_2·a_5_10
113.  − a_4_4·b_4_7 + b_2_0·a_6_7 + b_2_0·a_3_1·a_3_5
114. b_2_2·a_6_7 − a_3_4·a_5_7 + a_3_3·a_5_8 + a_3_3·a_5_7 + a_3_3·a_5_6
115. a_2_1·a_6_7
116. b_2_0·a_6_10 − b_2_0·a_3_1·a_3_5 + b_2_0·a_1_0·a_5_7 − b_2_0²·a_1_0·a_3_5
117. b_2_2·a_6_10 + a_3_4·a_5_7
118. a_2_1·a_6_10
119. a_4_4·b_4_7 + b_2_0·a_6_9 + b_2_0·a_3_1·a_3_5 − b_2_0²·a_1_0·a_3_5
120. b_2_2·a_6_9 + a_3_4·a_5_7 + a_3_3·a_5_10 + a_3_3·a_5_7 − a_3_3·a_5_6
121. a_2_1·a_6_9
122. b_4_7² − b_2_2⁴ + b_2_0·b_6_12 − a_4_4·b_4_7 + b_2_0·a_3_1·a_3_5 + b_2_0²·a_1_0·a_3_5
123. b_2_2·b_6_12 + a_4_4·b_4_7 − a_3_3·a_5_10 + a_3_3·a_5_8 + a_3_3·a_5_7 − a_3_3·a_5_6
124. a_2_1·b_6_12
125. a_1_1·a_7_17
126.  − a_4_4·b_4_7 + a_3_3·a_5_6 + a_1_0·a_7_17 + b_2_0·a_1_0·a_5_7 + b_2_0²·a_1_0·a_3_5
127. a_3_4·a_5_7 + a_1_1·a_7_18
128. a_4_4·b_4_7 + a_3_3·a_5_7 + a_3_3·a_5_6 + a_1_0·a_7_18 + b_2_0²·a_1_0·a_3_5
129. a_4_4·a_5_6
130. a_4_6·a_5_6
131. a_4_4·a_5_7 + a_1_0·a_3_1·a_5_6 + b_2_0·a_1_0·a_3_1·a_3_5
132.  − b_2_2²·a_5_6 + a_4_6·a_5_7 + a_1_0·a_3_1·a_5_6 + b_2_0·a_1_0·a_3_1·a_3_5
133.  − b_2_2²·a_5_6 + a_4_4·a_5_8 − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
134. b_2_2²·a_5_6 + a_4_6·a_5_8 − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
135. b_4_7·a_5_8 + b_4_7·a_5_7 + b_4_7·a_5_6 − b_2_2²·a_5_8 − b_2_2²·a_5_6 + b_2_2³·a_3_2
        + b_2_0²·a_5_9 − b_2_0²·a_5_6 + b_2_0³·a_3_5 + b_2_0³·a_3_3 + a_1_0·a_3_1·a_5_6
        − b_2_0·a_1_0·a_3_1·a_3_5
136.  − b_2_2²·a_5_6 + a_4_4·a_5_9
137. a_4_6·a_5_9
138.  − b_2_2²·a_5_6 + a_4_4·a_5_10 − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
139. b_4_7·a_5_10 + b_4_7·a_5_8 − b_4_7·a_5_7 − b_2_2²·a_5_10 − b_2_2²·a_5_8 − b_2_2³·a_3_2
        − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
140. a_4_6·a_5_10 − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
141.  − b_4_7·a_5_9 + b_4_7·a_5_8 + b_4_7·a_5_7 − b_4_7·a_5_6 − b_2_2²·a_5_8 + b_2_2³·a_3_2
        − b_2_0²·a_5_6 − b_2_0³·a_3_5 + b_2_0³·a_3_3 + b_2_0·a_1_0·a_3_1·a_3_5
        + b_2_0·c_6_14·a_1_0
142. a_6_7·a_3_3 + a_1_0·a_3_1·a_5_6 + b_2_0·a_1_0·a_3_1·a_3_5
143. a_6_7·a_3_1 + a_1_0·a_3_1·a_5_6 + b_2_0·a_1_0·a_3_1·a_3_5
144. a_6_7·a_3_5
145. b_2_2²·a_5_6 + a_6_7·a_3_4
146. a_6_7·a_3_2
147. a_6_10·a_3_3 − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
148. b_2_2²·a_5_6 + a_6_10·a_3_1 − b_2_0·a_1_0·a_3_1·a_3_5
149. a_6_10·a_3_5 + a_1_0·a_3_1·a_5_6 + b_2_0·a_1_0·a_3_1·a_3_5
150. b_2_2²·a_5_6 + a_6_10·a_3_4
151. a_6_10·a_3_2
152. a_6_9·a_3_3 + a_1_0·a_3_1·a_5_6 + b_2_0·a_1_0·a_3_1·a_3_5
153.  − b_2_2²·a_5_6 + a_6_9·a_3_1 − a_1_0·a_3_1·a_5_6
154. a_6_9·a_3_5
155. a_6_9·a_3_4
156. b_2_2²·a_5_6 + a_6_9·a_3_2
157. b_6_12·a_3_3 − b_4_7·a_5_9 + b_4_7·a_5_6 + b_2_2²·a_5_6 − b_2_0²·a_5_6 − b_2_0³·a_3_5
        − b_2_0³·a_3_3 + b_2_0·a_1_0·a_3_1·a_3_5
158. b_6_12·a_3_1 + b_4_7·a_5_7 + b_2_2³·a_3_2 + b_2_0²·a_5_6 + b_2_0³·a_3_5 + b_2_0³·a_3_3
        − a_1_0·a_3_1·a_5_6
159. b_6_12·a_3_5 + b_4_7·a_5_9 + b_2_2²·a_5_6 − b_2_0²·a_5_6 − b_2_0³·a_3_5 − b_2_0³·a_3_3
        + b_2_0·a_1_0·a_3_1·a_3_5
160. b_6_12·a_3_4
161. b_6_12·a_3_2
162. b_4_7·a_5_8 + b_4_7·a_5_7 − b_4_7·a_5_6 − b_2_2²·a_5_8 + b_2_2²·a_5_6 + b_2_2³·a_3_2
        + b_2_0·a_7_17 + b_2_0²·a_5_7 − b_2_0²·a_5_6 − b_2_0³·a_3_3 + a_1_0·a_3_1·a_5_6
        − b_2_0·a_1_0·a_3_1·a_3_5
163. a_2_1·a_7_17
164. b_4_7·a_5_8 − b_4_7·a_5_7 − b_4_7·a_5_6 − b_2_2²·a_5_8 + b_2_2²·a_5_6 − b_2_2³·a_3_2
        + b_2_0·a_7_18 + b_2_0²·a_5_6 − b_2_0³·a_3_5 − b_2_0³·a_3_3 + a_1_0·a_3_1·a_5_6
        − b_2_0·a_1_0·a_3_1·a_3_5
165. b_2_2·a_7_18 − b_2_2·a_7_17 + b_2_2²·a_5_10 − b_2_2²·a_5_6 − b_2_2³·a_3_2
        + a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
166. b_2_2²·a_5_6 + a_2_1·a_7_18
167. a_8_15·a_1_1
168.  − b_2_2²·a_5_6 + a_8_15·a_1_0 − a_1_0·a_3_1·a_5_6 − b_2_0·a_1_0·a_3_1·a_3_5
169. a_5_6·a_5_10 + a_5_6·a_5_8 − a_5_6·a_5_7
170. a_5_9·a_5_10 + a_5_7·a_5_10 − a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_6·a_5_9 + a_5_6·a_5_8
        + a_5_6·a_5_7 + b_2_2·a_3_2·a_5_10 + b_2_2·a_3_2·a_5_8 + b_2_0²·a_1_0·a_5_6
        + b_2_0³·a_1_0·a_3_5
171. a_5_7·a_5_9 − a_5_7·a_5_8 − a_5_6·a_5_8 + a_5_6·a_5_7 − b_2_2·a_3_2·a_5_8
        + b_2_0·a_3_1·a_5_6 + b_2_0²·a_3_1·a_3_5 + b_2_0²·a_1_0·a_5_7 − b_2_0²·a_1_0·a_5_6
        + c_6_14·a_1_0·a_3_1
172. a_5_6·a_5_9 + a_5_6·a_5_8 + a_5_6·a_5_7 + b_2_0²·a_1_0·a_5_6 + b_2_0³·a_1_0·a_3_5
        + c_6_14·a_1_0·a_3_5
173. a_5_8·a_5_9 + a_5_7·a_5_9 + a_5_6·a_5_9 + a_5_6·a_5_8 + a_5_6·a_5_7 + b_2_0²·a_1_0·a_5_6
        + b_2_0³·a_1_0·a_3_5 + c_6_14·a_1_0·a_3_2
174.  − a_5_9·a_5_10 + a_5_8·a_5_9 − a_5_7·a_5_9 + c_6_15·a_1_0·a_3_2
175. a_4_4·a_6_7
176. b_4_7·a_6_7 + a_5_9·a_5_10 + a_5_8·a_5_9 − a_5_7·a_5_8 − a_5_6·a_5_9 + a_5_6·a_5_7
        + b_2_0·a_3_1·a_5_6 + b_2_0²·a_3_1·a_3_5 − b_2_0²·a_1_0·a_5_7 − b_2_0²·a_1_0·a_5_6
        + b_2_0³·a_1_0·a_3_5
177. a_4_6·a_6_7
178. a_4_4·a_6_10
179. b_4_7·a_6_10 − b_2_0·a_3_1·a_5_6 − b_2_0²·a_3_1·a_3_5 + b_2_0²·a_1_0·a_5_7
        + b_2_0²·a_1_0·a_5_6 − b_2_0³·a_1_0·a_3_5
180. a_4_6·a_6_10
181. a_4_4·a_6_9
182. b_4_7·a_6_9 + a_5_8·a_5_9 − a_5_7·a_5_10 + a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_6·a_5_9
        + a_5_6·a_5_7 + b_2_2·a_3_2·a_5_8 + b_2_0·a_3_1·a_5_6 + b_2_0²·a_3_1·a_3_5
        − b_2_0²·a_1_0·a_5_7 − b_2_0²·a_1_0·a_5_6 + b_2_0³·a_1_0·a_3_5
183. a_4_6·a_6_9
184. a_4_4·b_6_12 − a_5_6·a_5_8 − a_5_6·a_5_7 + b_2_0²·a_1_0·a_5_6 + b_2_0³·a_1_0·a_3_5
185. b_4_7·b_6_12 + b_2_0²·b_6_12 + b_2_0³·b_4_7 − a_5_9·a_5_10 + a_5_8·a_5_9 + a_5_7·a_5_10
        − a_5_7·a_5_9 + a_5_6·a_5_8 − a_5_6·a_5_7 + b_2_2·a_3_2·a_5_8 + b_2_0·a_3_1·a_5_6
        − b_2_0²·a_3_1·a_3_5 + b_2_0²·a_1_0·a_5_7 − b_2_0²·a_1_0·a_5_6 − b_2_0³·a_1_0·a_3_5
        − b_2_0²·c_6_14
186. a_4_6·b_6_12 − a_5_6·a_5_8 − a_5_6·a_5_7 + b_2_0²·a_1_0·a_5_6 + b_2_0³·a_1_0·a_3_5
187.  − a_5_9·a_5_10 + a_5_8·a_5_9 + a_5_7·a_5_10 − a_5_7·a_5_9 + a_5_6·a_5_9 + a_5_6·a_5_8
        − a_5_6·a_5_7 + a_3_3·a_7_17 + b_2_2·a_3_2·a_5_8
188.  − a_5_6·a_5_8 − a_5_6·a_5_7 + b_2_0·a_1_0·a_7_17 + b_2_0²·a_1_0·a_5_7
        − b_2_0²·a_1_0·a_5_6
189. a_5_9·a_5_10 + a_5_8·a_5_9 + a_5_7·a_5_8 − a_5_6·a_5_9 − a_5_6·a_5_8 + a_3_1·a_7_17
        + b_2_2·a_3_2·a_5_8 − b_2_0·a_3_1·a_5_6 − b_2_0²·a_3_1·a_3_5 − b_2_0²·a_1_0·a_5_7
        + b_2_0²·a_1_0·a_5_6
190. a_5_9·a_5_10 − a_5_8·a_5_9 − a_5_7·a_5_10 + a_5_7·a_5_9 − a_5_6·a_5_9 − a_5_6·a_5_7
        + a_3_5·a_7_17 − b_2_2·a_3_2·a_5_8 − b_2_0·a_3_1·a_5_6 − b_2_0²·a_3_1·a_3_5
        + b_2_0²·a_1_0·a_5_7 + b_2_0³·a_1_0·a_3_5
191.  − a_5_9·a_5_10 − a_5_8·a_5_9 + a_5_6·a_5_9 + a_5_6·a_5_8 + a_5_6·a_5_7 + a_3_4·a_7_17
        + b_2_0²·a_1_0·a_5_6 + b_2_0³·a_1_0·a_3_5
192. a_5_9·a_5_10 + a_5_7·a_5_10 − a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_6·a_5_9 + a_5_6·a_5_8
        + a_5_6·a_5_7 + a_3_2·a_7_17 − b_2_2·a_3_2·a_5_8 + b_2_0²·a_1_0·a_5_6
        + b_2_0³·a_1_0·a_3_5
193. a_5_7·a_5_9 + a_5_6·a_5_9 − a_5_6·a_5_8 + a_3_3·a_7_18 + b_2_2·a_3_2·a_5_8
        + b_2_0·a_3_1·a_5_6 + b_2_0²·a_3_1·a_3_5 − b_2_0²·a_1_0·a_5_7 − b_2_0²·a_1_0·a_5_6
        + b_2_0³·a_1_0·a_3_5
194. a_5_9·a_5_10 + a_5_8·a_5_9 − a_5_7·a_5_8 − a_5_6·a_5_9 + a_5_6·a_5_8 − a_5_6·a_5_7
        − b_2_2·a_3_2·a_5_8 + b_2_0·a_1_0·a_7_18 + b_2_0³·a_1_0·a_3_5
195.  − a_5_8·a_5_9 − a_5_7·a_5_9 + a_5_7·a_5_8 − a_5_6·a_5_9 − a_5_6·a_5_8 + a_3_1·a_7_18
        + b_2_2·a_3_2·a_5_8 + b_2_0²·a_3_1·a_3_5 − b_2_0²·a_1_0·a_5_7 + b_2_0²·a_1_0·a_5_6
196.  − a_5_7·a_5_9 − a_5_6·a_5_9 − a_5_6·a_5_8 − a_5_6·a_5_7 + a_3_5·a_7_18 − b_2_2·a_3_2·a_5_8
        + b_2_0·a_3_1·a_5_6 + b_2_0²·a_3_1·a_3_5 − b_2_0²·a_1_0·a_5_7 − b_2_0³·a_1_0·a_3_5
197. a_5_9·a_5_10 + a_5_8·a_5_9 − a_5_6·a_5_9 − a_5_6·a_5_8 − a_5_6·a_5_7 + a_3_4·a_7_18
        − b_2_0²·a_1_0·a_5_6 − b_2_0³·a_1_0·a_3_5
198.  − a_5_9·a_5_10 − a_5_8·a_5_9 + a_5_6·a_5_9 + a_5_6·a_5_8 + a_5_6·a_5_7 + a_3_2·a_7_18
        + b_2_2·a_3_2·a_5_8 + b_2_0²·a_1_0·a_5_6 + b_2_0³·a_1_0·a_3_5
199. b_2_0·a_8_15 − a_5_9·a_5_10 − a_5_8·a_5_9 + a_5_7·a_5_8 + a_5_6·a_5_9 − a_5_6·a_5_8
        + a_5_6·a_5_7 + b_2_2·a_3_2·a_5_8 − b_2_0·a_3_1·a_5_6 − b_2_0²·a_3_1·a_3_5
        − b_2_0²·a_1_0·a_5_7 + b_2_0³·a_1_0·a_3_5
200. b_2_2·a_8_15 + a_5_9·a_5_10 − a_5_8·a_5_10 + a_5_8·a_5_9 − a_5_7·a_5_10 + a_5_7·a_5_8
        − a_5_6·a_5_9 − a_5_6·a_5_8 − a_5_6·a_5_7 − b_2_0²·a_1_0·a_5_6 − b_2_0³·a_1_0·a_3_5
201. a_2_1·a_8_15
202.  − a_6_7·a_5_7 + a_6_7·a_5_6
203. a_6_7·a_5_9 + b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0²·a_1_0·a_3_1·a_3_5
204. a_6_7·a_5_8 − a_6_7·a_5_7 + b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0²·a_1_0·a_3_1·a_3_5
205. a_6_10·a_5_7 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
206. a_6_10·a_5_6
207. a_6_10·a_5_9 − a_6_7·a_5_7 + b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0²·a_1_0·a_3_1·a_3_5
208. a_6_10·a_5_10 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
209. a_6_10·a_5_8
210. a_6_9·a_5_7 + a_6_7·a_5_7 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
211. a_6_9·a_5_6 − a_6_7·a_5_7
212. a_6_9·a_5_9 + b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0²·a_1_0·a_3_1·a_3_5
213. a_6_9·a_5_10 + a_6_7·a_5_7
214. a_6_9·a_5_8 + a_6_7·a_5_10 + b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0²·a_1_0·a_3_1·a_3_5
215. b_6_12·a_5_9 − b_2_0³·a_5_9 − b_2_0³·a_5_6 + b_2_0⁴·a_3_3 + b_2_0²·a_1_0·a_3_1·a_3_5
        − b_2_0·c_6_14·a_3_5 − b_2_0²·c_6_14·a_1_0
216. b_6_12·a_5_10 + b_6_12·a_5_7 − b_6_12·a_5_6 − b_2_0³·a_5_9 + b_2_0³·a_5_6
        − b_2_0⁴·a_3_5 + b_2_0⁴·a_3_3 − a_6_7·a_5_10 + a_6_7·a_5_7 − b_2_0·a_1_0·a_3_1·a_5_6
        + b_2_0²·a_1_0·a_3_1·a_3_5 + b_2_0²·c_6_14·a_1_0
217. b_6_12·a_5_8 + b_6_12·a_5_7 + b_6_12·a_5_6 + b_2_0³·a_5_9 − b_2_0³·a_5_6 + b_2_0⁴·a_3_5
        − b_2_0⁴·a_3_3 − a_6_7·a_5_10 + a_6_7·a_5_7 + b_2_0·a_1_0·a_3_1·a_5_6
        − b_2_0²·a_1_0·a_3_1·a_3_5 − b_2_0²·c_6_14·a_1_0
218.  − b_6_12·a_5_6 + b_2_0²·a_7_17 − b_2_0³·a_5_9 + b_2_0³·a_5_7 + b_2_0³·a_5_6
        − b_2_0⁴·a_3_3 − a_6_7·a_5_7 + b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0·c_6_14·a_3_5
        + b_2_0·c_6_14·a_3_3
219.  − a_6_7·a_5_7 + a_1_0·a_3_1·a_7_17 − b_2_0·a_1_0·a_3_1·a_5_6
        − b_2_0²·a_1_0·a_3_1·a_3_5
220. b_2_2²·a_7_17 − b_2_2³·a_5_10 + b_2_2³·a_5_8 + b_2_2⁴·a_3_2 + a_6_7·a_5_10
        − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
221. a_4_4·a_7_17 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
222. b_6_12·a_5_7 + b_6_12·a_5_6 + b_4_7·a_7_17 − b_2_2³·a_5_10 + b_2_2³·a_5_8
        + b_2_2⁴·a_3_2 − b_2_0³·a_5_9 + b_2_0³·a_5_7 − b_2_0⁴·a_3_5 + a_6_7·a_5_10
        + b_2_0·c_6_14·a_3_5 + b_2_0·c_6_14·a_3_3 − b_2_0·c_6_14·a_3_1 − b_2_0²·c_6_14·a_1_0
223. a_4_6·a_7_17 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
224. b_6_12·a_5_7 + b_2_0²·a_7_18 − b_2_0³·a_5_9 + b_2_0³·a_5_7 − b_2_0·a_1_0·a_3_1·a_5_6
        + b_2_0²·a_1_0·a_3_1·a_3_5 − b_2_0·c_6_14·a_3_1
225. a_6_7·a_5_7 + a_4_4·a_7_18 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
226.  − b_6_12·a_5_7 + b_6_12·a_5_6 + b_4_7·a_7_18 + b_2_2³·a_5_8 + b_2_0³·a_5_9
        − b_2_0³·a_5_6 + b_2_0⁴·a_3_5 − b_2_0⁴·a_3_3 + a_6_7·a_5_10 + a_6_7·a_5_7
        − b_2_0·a_1_0·a_3_1·a_5_6 + b_2_0·c_6_14·a_3_5 + b_2_0·c_6_14·a_3_3
        − b_2_0²·c_6_14·a_1_0
227. a_6_7·a_5_7 + a_4_6·a_7_18 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
228. a_8_15·a_3_3 + a_6_7·a_5_10 + a_6_7·a_5_7 + b_2_0·a_1_0·a_3_1·a_5_6
        + b_2_0²·a_1_0·a_3_1·a_3_5
229. a_8_15·a_3_1 − a_6_7·a_5_7 + b_2_0²·a_1_0·a_3_1·a_3_5
230. a_8_15·a_3_5 − a_6_7·a_5_10 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
231. a_8_15·a_3_4
232. a_8_15·a_3_2 + a_6_7·a_5_10 − b_2_0·a_1_0·a_3_1·a_5_6 − b_2_0²·a_1_0·a_3_1·a_3_5
233. a_6_7²
234. a_6_10²
235. a_6_7·a_6_10
236. a_6_10·a_6_9
237. a_6_9²
238. a_6_7·a_6_9
239.  − a_6_9·b_6_12 + a_6_7·b_6_12 − b_2_0³·a_1_0·a_5_6 − b_2_0⁴·a_1_0·a_3_5
        + b_2_0·c_6_14·a_1_0·a_3_5
240. b_6_12² − b_2_0³·b_6_12 + b_2_0⁴·b_4_7 + a_6_9·b_6_12 + a_5_7·a_7_17
        + b_2_2²·a_3_2·a_5_10 − b_2_2²·a_3_2·a_5_8 − b_2_0²·a_3_1·a_5_6
        + b_2_0³·a_3_1·a_3_5 − b_2_0³·a_1_0·a_5_7 − b_2_0³·a_1_0·a_5_6 + b_2_0⁴·a_1_0·a_3_5
        + b_2_0·b_4_7·c_6_14 − b_2_0³·c_6_14 − c_6_15·a_1_0·a_5_8 − c_6_15·a_1_0·a_5_7
        − c_6_15·a_1_0·a_5_6 − c_6_14·a_3_1·a_3_5 − c_6_14·a_1_0·a_5_8 − c_6_14·a_1_0·a_5_6
        + b_2_0·c_6_14·a_1_0·a_3_1
241. b_6_12² − b_2_0³·b_6_12 + b_2_0⁴·b_4_7 + a_5_6·a_7_17 + b_2_0²·a_3_1·a_5_6
        − b_2_0³·a_1_0·a_5_7 − b_2_0³·a_1_0·a_5_6 + b_2_0⁴·a_1_0·a_3_5 + b_2_0·b_4_7·c_6_14
        − b_2_0³·c_6_14 − b_2_0·c_6_14·a_1_0·a_3_5 − b_2_0·c_6_14·a_1_0·a_3_1
242.  − b_6_12² + b_2_0³·b_6_12 − b_2_0⁴·b_4_7 + a_6_10·b_6_12 + b_2_0²·a_1_0·a_7_17
        + b_2_0³·a_3_1·a_3_5 + b_2_0³·a_1_0·a_5_7 + b_2_0⁴·a_1_0·a_3_5 − b_2_0·b_4_7·c_6_14
        + b_2_0³·c_6_14 − b_2_0·c_6_14·a_1_0·a_3_5 + b_2_0·c_6_14·a_1_0·a_3_1
243. b_6_12² − b_2_0³·b_6_12 + b_2_0⁴·b_4_7 + a_6_9·b_6_12 − a_6_10·b_6_12
        + b_2_0·a_3_1·a_7_17 + b_2_0²·a_3_1·a_5_6 − b_2_0³·a_1_0·a_5_7 − b_2_0⁴·a_1_0·a_3_5
        + b_2_0·b_4_7·c_6_14 − b_2_0³·c_6_14 − b_2_0·c_6_14·a_1_0·a_3_5
244.  − b_6_12² + b_2_0³·b_6_12 − b_2_0⁴·b_4_7 − a_6_9·b_6_12 + a_6_10·b_6_12 + a_5_9·a_7_17
        − b_2_0²·a_3_1·a_5_6 + b_2_0³·a_1_0·a_5_7 + b_2_0⁴·a_1_0·a_3_5 − b_2_0·b_4_7·c_6_14
        + b_2_0³·c_6_14 + c_6_15·a_1_0·a_5_10 − c_6_15·a_1_0·a_5_8 + c_6_15·a_1_0·a_5_6
        + c_6_14·a_1_0·a_5_10 − c_6_14·a_1_0·a_5_8 − c_6_14·a_1_0·a_5_6
        + b_2_0·c_6_14·a_1_0·a_3_1
245.  − b_6_12² + b_2_0³·b_6_12 − b_2_0⁴·b_4_7 − a_6_10·b_6_12 + a_5_10·a_7_17
        − b_2_2·a_5_8·a_5_10 − b_2_2²·a_3_2·a_5_10 − b_2_0²·a_3_1·a_5_6 − b_2_0³·a_1_0·a_5_7
        − b_2_0⁴·a_1_0·a_3_5 − b_2_0·b_4_7·c_6_14 + b_2_0³·c_6_14 + c_6_15·a_1_0·a_5_10
        − c_6_15·a_1_0·a_5_8 + c_6_15·a_1_0·a_5_6 + c_6_14·a_3_1·a_3_5 + c_6_14·a_1_0·a_5_10
        − c_6_14·a_1_0·a_5_8 − c_6_14·a_1_0·a_5_7 + c_6_14·a_1_0·a_5_6
246.  − b_6_12² + b_2_0³·b_6_12 − b_2_0⁴·b_4_7 + a_6_9·b_6_12 + a_6_10·b_6_12 + a_5_8·a_7_17
        − b_2_2·a_5_8·a_5_10 − b_2_2²·a_3_2·a_5_8 + b_2_0³·a_3_1·a_3_5 − b_2_0³·a_1_0·a_5_6
        − b_2_0⁴·a_1_0·a_3_5 − b_2_0·b_4_7·c_6_14 + b_2_0³·c_6_14 + c_6_15·a_1_0·a_5_10
        + c_6_15·a_1_0·a_5_7 − c_6_15·a_1_0·a_5_6 + c_6_14·a_3_1·a_3_5 + c_6_14·a_1_0·a_5_10
        − c_6_14·a_1_0·a_5_6 + b_2_0·c_6_14·a_1_0·a_3_1
247. b_6_12² − b_2_0³·b_6_12 + b_2_0⁴·b_4_7 − a_6_9·b_6_12 − a_6_10·b_6_12 + a_5_7·a_7_18
        − b_2_2²·a_3_2·a_5_8 − b_2_0²·a_3_1·a_5_6 + b_2_0³·a_3_1·a_3_5 + b_2_0³·a_1_0·a_5_7
        − b_2_0³·a_1_0·a_5_6 + b_2_0·b_4_7·c_6_14 − b_2_0³·c_6_14 − c_6_15·a_1_0·a_5_10
        − c_6_15·a_1_0·a_5_8 + c_6_15·a_1_0·a_5_7 − c_6_14·a_3_1·a_3_5 − c_6_14·a_1_0·a_5_10
        + c_6_14·a_1_0·a_5_6 + b_2_0·c_6_14·a_1_0·a_3_5 − b_2_0·c_6_14·a_1_0·a_3_1
248.  − b_6_12² + b_2_0³·b_6_12 − b_2_0⁴·b_4_7 − a_6_10·b_6_12 + a_5_6·a_7_18
        − b_2_0²·a_3_1·a_5_6 + b_2_0³·a_1_0·a_5_7 + b_2_0³·a_1_0·a_5_6 − b_2_0⁴·a_1_0·a_3_5
        − b_2_0·b_4_7·c_6_14 + b_2_0³·c_6_14 + c_6_14·a_3_1·a_3_5 − c_6_14·a_1_0·a_5_7
        − b_2_0·c_6_14·a_1_0·a_3_5 + b_2_0·c_6_14·a_1_0·a_3_1
249.  − a_6_9·b_6_12 − a_6_10·b_6_12 + b_2_0²·a_1_0·a_7_18 + b_2_0³·a_1_0·a_5_7
        − b_2_0⁴·a_1_0·a_3_5 − b_2_0·c_6_14·a_1_0·a_3_5 − b_2_0·c_6_14·a_1_0·a_3_1
250.  − b_6_12² + b_2_0³·b_6_12 − b_2_0⁴·b_4_7 − a_6_9·b_6_12 + a_5_9·a_7_18
        − b_2_0²·a_3_1·a_5_6 + b_2_0³·a_1_0·a_5_6 + b_2_0⁴·a_1_0·a_3_5 − b_2_0·b_4_7·c_6_14
        + b_2_0³·c_6_14 − c_6_15·a_1_0·a_5_10 + c_6_15·a_1_0·a_5_8 − c_6_15·a_1_0·a_5_6
        + c_6_14·a_3_1·a_3_5 + c_6_14·a_1_0·a_5_10 − c_6_14·a_1_0·a_5_8 − c_6_14·a_1_0·a_5_6
        + b_2_0·c_6_14·a_1_0·a_3_5 − b_2_0·c_6_14·a_1_0·a_3_1
251. a_5_10·a_7_18 − b_2_2·a_5_8·a_5_10 + c_6_15·a_1_0·a_5_10 − c_6_15·a_1_0·a_5_8
        + c_6_15·a_1_0·a_5_6 − c_6_14·a_3_1·a_3_5 + c_6_14·a_1_0·a_5_8 − c_6_14·a_1_0·a_5_7
        + c_6_14·a_1_0·a_5_6
252. b_6_12² − b_2_0³·b_6_12 + b_2_0⁴·b_4_7 − a_6_9·b_6_12 − a_6_10·b_6_12 + a_5_8·a_7_18
        − b_2_0²·a_3_1·a_5_6 + b_2_0³·a_3_1·a_3_5 + b_2_0³·a_1_0·a_5_7 − b_2_0³·a_1_0·a_5_6
        + b_2_0·b_4_7·c_6_14 − b_2_0³·c_6_14 − c_6_15·a_1_0·a_5_8 − c_6_15·a_1_0·a_5_7
        − c_6_15·a_1_0·a_5_6 − c_6_14·a_1_0·a_5_10 − c_6_14·a_1_0·a_5_8 + c_6_14·a_1_0·a_5_7
        + b_2_0·c_6_14·a_1_0·a_3_5 − b_2_0·c_6_14·a_1_0·a_3_1
253. a_4_4·a_8_15
254. b_6_12² − b_2_0³·b_6_12 + b_2_0⁴·b_4_7 + a_6_9·b_6_12 + a_6_10·b_6_12 + b_4_7·a_8_15
        − b_2_2·a_5_8·a_5_10 + b_2_2²·a_3_2·a_5_10 − b_2_2²·a_3_2·a_5_8 − b_2_0³·a_3_1·a_3_5
        + b_2_0³·a_1_0·a_5_7 + b_2_0³·a_1_0·a_5_6 − b_2_0⁴·a_1_0·a_3_5 + b_2_0·b_4_7·c_6_14
        − b_2_0³·c_6_14 + b_2_0·c_6_14·a_1_0·a_3_1
255. a_4_6·a_8_15
256. a_6_10·a_7_17 − b_2_0·a_1_0·a_3_1·a_7_17 − b_2_0²·a_1_0·a_3_1·a_5_6
        − b_2_0³·a_1_0·a_3_1·a_3_5 + a_2_1·c_6_15·a_5_7 + a_2_1·c_6_14·a_5_7
257. a_6_9·a_7_17 + a_3_2·a_5_8·a_5_10 − b_2_0²·a_1_0·a_3_1·a_5_6
        − b_2_0³·a_1_0·a_3_1·a_3_5 + c_6_14·a_1_0·a_3_1·a_3_5
258. a_6_7·a_7_17 + a_3_2·a_5_8·a_5_10 − b_2_0·a_1_0·a_3_1·a_7_17
        − b_2_0²·a_1_0·a_3_1·a_5_6 − b_2_0³·a_1_0·a_3_1·a_3_5 + a_2_1·c_6_15·a_5_7
        + a_2_1·c_6_14·a_5_7 + c_6_14·a_1_0·a_3_1·a_3_5
259.  − b_6_12·a_7_17 + b_2_0³·a_7_18 + b_2_0³·a_7_17 + b_2_0⁴·a_5_9 − b_2_0⁴·a_5_7
        − b_2_0⁴·a_5_6 + b_2_0⁵·a_3_5 + b_2_0⁵·a_3_3 + b_2_0·a_1_0·a_3_1·a_7_17
        − b_2_0·c_6_14·a_5_6 − b_2_0²·c_6_14·a_3_5 + b_2_0²·c_6_14·a_3_3
        − b_2_0²·c_6_14·a_3_1 − b_2_0³·c_6_14·a_1_0 − c_6_14·a_1_0·a_3_1·a_3_5
260. b_6_12·a_7_18 − b_6_12·a_7_17 − b_2_0⁴·a_5_9 − b_2_0⁴·a_5_7 − b_2_0⁵·a_3_3
        − a_3_2·a_5_8·a_5_10 + b_2_0²·a_1_0·a_3_1·a_5_6 + b_2_0³·a_1_0·a_3_1·a_3_5
        + b_2_0·c_6_14·a_5_7 − b_2_0²·c_6_14·a_3_5 − b_2_0²·c_6_14·a_3_3
        + b_2_0²·c_6_14·a_3_1 + b_2_0³·c_6_14·a_1_0 + c_6_14·a_1_0·a_3_1·a_3_5
261. a_6_10·a_7_18 + b_2_0·a_1_0·a_3_1·a_7_17 + b_2_0²·a_1_0·a_3_1·a_5_6
        + b_2_0³·a_1_0·a_3_1·a_3_5 − a_2_1·c_6_15·a_5_7 − a_2_1·c_6_14·a_5_7
262. a_6_9·a_7_18 + a_3_2·a_5_8·a_5_10 + b_2_0·a_1_0·a_3_1·a_7_17 − a_2_1·c_6_14·a_5_7
263. a_6_7·a_7_18 + b_2_0·a_1_0·a_3_1·a_7_17 + b_2_0²·a_1_0·a_3_1·a_5_6
        + b_2_0³·a_1_0·a_3_1·a_3_5 − a_2_1·c_6_15·a_5_7 + a_2_1·c_6_14·a_5_7
        − c_6_14·a_1_0·a_3_1·a_3_5
264. a_8_15·a_5_7 + a_3_2·a_5_8·a_5_10 + b_2_0·a_1_0·a_3_1·a_7_17
        − b_2_0²·a_1_0·a_3_1·a_5_6 − b_2_0³·a_1_0·a_3_1·a_3_5 − c_6_14·a_1_0·a_3_1·a_3_5
265. a_8_15·a_5_6 + b_2_0²·a_1_0·a_3_1·a_5_6 + b_2_0³·a_1_0·a_3_1·a_3_5
        + c_6_14·a_1_0·a_3_1·a_3_5
266. a_8_15·a_5_9 + b_2_0²·a_1_0·a_3_1·a_5_6 + b_2_0³·a_1_0·a_3_1·a_3_5
        − a_2_1·c_6_14·a_5_7
267. a_8_15·a_5_10 − a_3_2·a_5_8·a_5_10 + b_2_0·a_1_0·a_3_1·a_7_17
        + b_2_0²·a_1_0·a_3_1·a_5_6 + b_2_0³·a_1_0·a_3_1·a_3_5 + a_2_1·c_6_14·a_5_7
        − c_6_14·a_1_0·a_3_1·a_3_5
268. a_8_15·a_5_8 − a_3_2·a_5_8·a_5_10 + b_2_0²·a_1_0·a_3_1·a_5_6
        + b_2_0³·a_1_0·a_3_1·a_3_5 + a_2_1·c_6_14·a_5_7
269. a_7_17·a_7_18 − b_2_2²·a_5_8·a_5_10 − b_2_2³·a_3_2·a_5_8 + b_2_0³·a_3_1·a_5_6
        + b_2_0³·a_1_0·a_7_18 + b_2_0³·a_1_0·a_7_17 + b_2_0⁴·a_3_1·a_3_5
        + b_2_0⁴·a_1_0·a_5_6 − b_2_0⁵·a_1_0·a_3_5 + c_6_15·a_1_1·a_7_18 − c_6_14·a_3_1·a_5_6
        + c_6_14·a_1_1·a_7_18 + b_2_0·c_6_14·a_1_0·a_5_7 + b_2_0²·c_6_14·a_1_0·a_3_5
270. b_6_12·a_8_15 + b_2_0²·a_3_1·a_7_17 − b_2_0³·a_3_1·a_5_6 + b_2_0³·a_1_0·a_7_18
        + b_2_0³·a_1_0·a_7_17 − b_2_0⁴·a_3_1·a_3_5 + b_2_0⁴·a_1_0·a_5_7
        + b_2_0⁴·a_1_0·a_5_6 − b_2_0·c_6_14·a_3_1·a_3_5 − b_2_0·c_6_14·a_1_0·a_5_6
        + b_2_0²·c_6_14·a_1_0·a_3_5 − b_2_0²·c_6_14·a_1_0·a_3_1
271. a_6_10·a_8_15
272. a_6_9·a_8_15
273. a_6_7·a_8_15
274. a_8_15·a_7_18 + b_2_2·a_3_2·a_5_8·a_5_10 − b_2_0²·a_1_0·a_3_1·a_7_17
        + b_2_0³·a_1_0·a_3_1·a_5_6 + b_2_0⁴·a_1_0·a_3_1·a_3_5
275. a_8_15·a_7_17 + b_2_2·a_3_2·a_5_8·a_5_10 − b_2_0²·a_1_0·a_3_1·a_7_17
        − c_6_14·a_1_0·a_3_1·a_5_6
276. a_8_15²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243

Data used for Benson′s test

• Benson′s completion test succeeded in degree 16.
• 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_6_14, a Duflot regular element of degree 6
  2. c_6_15, a Duflot regular element of degree 6
  3. b_2_2² + b_2_0², an element of degree 4
• The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 13].
• 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 243

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_1 → 0, an element of degree 2
4. b_2_0 → 0, an element of degree 2
5. b_2_2 → 0, an element of degree 2
6. a_3_1 → 0, an element of degree 3
7. a_3_2 → 0, an element of degree 3
8. a_3_3 → 0, an element of degree 3
9. a_3_4 → 0, an element of degree 3
10. a_3_5 → 0, an element of degree 3
11. a_4_4 → 0, an element of degree 4
12. a_4_6 → 0, an element of degree 4
13. b_4_7 → 0, an element of degree 4
14. a_5_6 → 0, an element of degree 5
15. a_5_7 → 0, an element of degree 5
16. a_5_8 → 0, an element of degree 5
17. a_5_9 → 0, an element of degree 5
18. a_5_10 → 0, an element of degree 5
19. a_6_7 → 0, an element of degree 6
20. a_6_10 → 0, an element of degree 6
21. a_6_9 → 0, an element of degree 6
22. b_6_12 → 0, an element of degree 6
23. c_6_14 → c_2_2³, an element of degree 6
24. c_6_15 →  − c_2_2³ + c_2_1³, an element of degree 6
25. a_7_17 → 0, an element of degree 7
26. a_7_18 → 0, an element of degree 7
27. a_8_15 → 0, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → a_1_2, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. b_2_0 → c_2_5, an element of degree 2
5. b_2_2 →  − a_1_0·a_1_2, an element of degree 2
6. a_3_1 → c_2_5·a_1_0 − c_2_3·a_1_2, an element of degree 3
7. a_3_2 → 0, an element of degree 3
8. a_3_3 →  − c_2_4·a_1_2, an element of degree 3
9. a_3_4 → 0, an element of degree 3
10. a_3_5 →  − c_2_5·a_1_1 + c_2_4·a_1_2, an element of degree 3
11. a_4_4 → c_2_5·a_1_1·a_1_2, an element of degree 4
12. a_4_6 → c_2_5·a_1_1·a_1_2, an element of degree 4
13. b_4_7 →  − c_2_4·c_2_5, an element of degree 4
14. a_5_6 →  − c_2_5·a_1_0·a_1_1·a_1_2 + c_2_5²·a_1_1 + c_2_4·c_2_5·a_1_1, an element of degree 5
15. a_5_7 → c_2_5·a_1_0·a_1_1·a_1_2 + c_2_5²·a_1_1 − c_2_4·c_2_5·a_1_0 + c_2_3·c_2_4·a_1_2, an element of degree 5
16. a_5_8 → c_2_5·a_1_0·a_1_1·a_1_2 + c_2_5²·a_1_1 − c_2_4·c_2_5·a_1_2 − c_2_4·c_2_5·a_1_1
       + c_2_4·c_2_5·a_1_0 − c_2_3·c_2_4·a_1_2, an element of degree 5
17. a_5_9 →  − c_2_5²·a_1_1 + c_2_4·c_2_5·a_1_1 − c_2_4²·a_1_2, an element of degree 5
18. a_5_10 →  − c_2_5·a_1_0·a_1_1·a_1_2 + c_2_4·c_2_5·a_1_2 + c_2_4·c_2_5·a_1_1 + c_2_4·c_2_5·a_1_0
       − c_2_3·c_2_4·a_1_2, an element of degree 5
19. a_6_7 → c_2_5²·a_1_0·a_1_1 − c_2_4·c_2_5·a_1_1·a_1_2 − c_2_4·c_2_5·a_1_0·a_1_2
       + c_2_3·c_2_5·a_1_1·a_1_2, an element of degree 6
20. a_6_10 →  − c_2_5²·a_1_1·a_1_2 − c_2_5²·a_1_0·a_1_1 − c_2_3·c_2_5·a_1_1·a_1_2, an element of degree 6
21. a_6_9 → c_2_5²·a_1_1·a_1_2 + c_2_5²·a_1_0·a_1_1 + c_2_4·c_2_5·a_1_1·a_1_2
       − c_2_4·c_2_5·a_1_0·a_1_2 + c_2_3·c_2_5·a_1_1·a_1_2, an element of degree 6
22. b_6_12 →  − c_2_5²·a_1_1·a_1_2 + c_2_5²·a_1_0·a_1_1 − c_2_4·c_2_5·a_1_1·a_1_2
       − c_2_4·c_2_5·a_1_0·a_1_2 + c_2_3·c_2_5·a_1_1·a_1_2 − c_2_4²·c_2_5, an element of degree 6
23. c_6_14 → c_2_5²·a_1_1·a_1_2 + c_2_4·c_2_5·a_1_1·a_1_2 − c_2_4·c_2_5·a_1_0·a_1_2
       − c_2_4·c_2_5² − c_2_4²·c_2_5 + c_2_4³, an element of degree 6
24. c_6_15 →  − c_2_5²·a_1_1·a_1_2 − c_2_5²·a_1_0·a_1_1 + c_2_4·c_2_5·a_1_1·a_1_2
       − c_2_3·c_2_5·a_1_1·a_1_2 − c_2_4·c_2_5² − c_2_4²·c_2_5 − c_2_4³ − c_2_3·c_2_5²
       + c_2_3³, an element of degree 6
25. a_7_17 →  − c_2_5²·a_1_0·a_1_1·a_1_2 − c_2_4·c_2_5·a_1_0·a_1_1·a_1_2 − c_2_4·c_2_5²·a_1_2
       − c_2_4·c_2_5²·a_1_1 + c_2_4·c_2_5²·a_1_0 − c_2_4²·c_2_5·a_1_2 + c_2_4²·c_2_5·a_1_1
       − c_2_3·c_2_4·c_2_5·a_1_2, an element of degree 7
26. a_7_18 →  − c_2_5²·a_1_0·a_1_1·a_1_2 + c_2_5³·a_1_1 + c_2_4·c_2_5²·a_1_1 − c_2_4²·c_2_5·a_1_2
       + c_2_4²·c_2_5·a_1_1 − c_2_4²·c_2_5·a_1_0 + c_2_3·c_2_4²·a_1_2, an element of degree 7
27. a_8_15 → c_2_5³·a_1_1·a_1_2 + c_2_4·c_2_5²·a_1_1·a_1_2 − c_2_4·c_2_5²·a_1_0·a_1_2
       + c_2_4·c_2_5²·a_1_0·a_1_1 + c_2_4²·c_2_5·a_1_1·a_1_2 − c_2_4²·c_2_5·a_1_0·a_1_2
       + c_2_3·c_2_4·c_2_5·a_1_1·a_1_2, 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_1 → 0, an element of degree 2
4. b_2_0 → 0, an element of degree 2
5. b_2_2 →  − c_2_5, an element of degree 2
6. a_3_1 → 0, an element of degree 3
7. a_3_2 → c_2_5·a_1_2, an element of degree 3
8. a_3_3 → c_2_5·a_1_2, an element of degree 3
9. a_3_4 → 0, an element of degree 3
10. a_3_5 →  − c_2_5·a_1_2, an element of degree 3
11. a_4_4 → 0, an element of degree 4
12. a_4_6 → 0, an element of degree 4
13. b_4_7 → c_2_5², an element of degree 4
14. a_5_6 → 0, an element of degree 5
15. a_5_7 → c_2_5²·a_1_2, an element of degree 5
16. a_5_8 → c_2_5²·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
17. a_5_9 → 0, an element of degree 5
18. a_5_10 →  − c_2_5²·a_1_2 + c_2_5²·a_1_1 − c_2_5²·a_1_0 − c_2_4·c_2_5·a_1_2 + c_2_3·c_2_5·a_1_2, an element of degree 5
19. a_6_7 →  − c_2_5²·a_1_1·a_1_2, an element of degree 6
20. a_6_10 → 0, an element of degree 6
21. a_6_9 →  − c_2_5²·a_1_1·a_1_2 + c_2_5²·a_1_0·a_1_2, an element of degree 6
22. b_6_12 →  − c_2_5²·a_1_0·a_1_2, an element of degree 6
23. c_6_14 → c_2_5²·a_1_0·a_1_2 + c_2_5³ − c_2_4·c_2_5² + c_2_4³, an element of degree 6
24. c_6_15 →  − c_2_5²·a_1_1·a_1_2 − c_2_5²·a_1_0·a_1_2 + c_2_4·c_2_5² − c_2_4³ − c_2_3·c_2_5²
       + c_2_3³, an element of degree 6
25. a_7_17 →  − c_2_5²·a_1_0·a_1_1·a_1_2 + c_2_5³·a_1_0 − c_2_3·c_2_5²·a_1_2, an element of degree 7
26. a_7_18 →  − c_2_5²·a_1_0·a_1_1·a_1_2 + c_2_5³·a_1_1 − c_2_4·c_2_5²·a_1_2, an element of degree 7
27. a_8_15 → c_2_5³·a_1_1·a_1_2 − c_2_5³·a_1_0·a_1_2 − c_2_5³·a_1_0·a_1_1
       + c_2_4·c_2_5²·a_1_0·a_1_2 − c_2_3·c_2_5²·a_1_1·a_1_2, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

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