Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 2053 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has 4 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) · (t8 + t7 + 2·t5 + t4 + t3 + t2 + t + 1) |
| (t − 1)3 · (t2 + 1)2 · (t4 + 1) |
- The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 8:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- b_3_10, an element of degree 3
- b_3_11, an element of degree 3
- c_4_16, a Duflot regular element of degree 4
- b_5_22, an element of degree 5
- b_5_23, an element of degree 5
- b_5_24, an element of degree 5
- b_7_40, an element of degree 7
- c_8_49, a Duflot regular element of degree 8
Ring relations
There are 36 minimal relations of maximal degree 14:
- b_1_0·b_1_2
- b_1_1·b_1_3 + b_1_0·b_1_1 + b_1_02
- b_1_2·b_1_32 + b_1_23 + b_1_0·b_1_32 + b_1_0·b_1_12 + b_1_03
- b_1_02·b_1_3 + b_1_02·b_1_1 + b_1_03
- b_1_0·b_3_10
- b_1_3·b_3_10 + b_1_2·b_3_11 + b_1_1·b_3_10 + b_1_0·b_1_33
- b_1_0·b_1_14 + b_1_05
- b_1_2·b_1_3·b_3_11 + b_1_22·b_3_10 + b_1_0·b_1_3·b_3_11 + b_1_0·b_1_1·b_3_11
+ b_1_02·b_3_11
- b_3_102 + b_1_12·b_1_2·b_3_10 + b_1_0·b_1_12·b_3_11 + b_1_03·b_3_11
+ c_4_16·b_1_22
- b_3_10·b_3_11 + b_1_12·b_1_2·b_3_10 + b_1_13·b_3_10 + b_1_0·b_1_32·b_3_11
+ c_4_16·b_1_2·b_1_3 + c_4_16·b_1_1·b_1_2
- b_3_10·b_3_11 + b_1_2·b_5_22 + b_1_23·b_3_11 + b_1_13·b_3_10 + b_1_0·b_1_12·b_3_11
+ b_1_03·b_3_11
- b_3_112 + b_1_13·b_3_11 + b_1_13·b_3_10 + b_1_0·b_5_22 + b_1_0·b_1_12·b_3_11
+ b_1_06 + c_4_16·b_1_32 + c_4_16·b_1_12 + c_4_16·b_1_0·b_1_3 + c_4_16·b_1_0·b_1_1
- b_1_3·b_5_22 + b_1_33·b_3_11 + b_1_0·b_5_23 + b_1_0·b_1_32·b_3_11 + b_1_0·b_1_35
+ b_1_0·b_1_12·b_3_11 + b_1_02·b_1_1·b_3_11 + b_1_05·b_1_1 + b_1_06 + c_4_16·b_1_32 + c_4_16·b_1_0·b_1_3 + c_4_16·b_1_0·b_1_1 + c_4_16·b_1_02
- b_3_112 + b_1_1·b_5_23 + b_1_1·b_5_22 + b_1_0·b_1_32·b_3_11 + b_1_05·b_1_1
+ c_4_16·b_1_32 + c_4_16·b_1_1·b_1_2
- b_3_102 + b_1_3·b_5_24 + b_1_3·b_5_23 + b_1_3·b_5_22 + b_1_33·b_3_11 + b_1_2·b_5_23
+ b_1_23·b_3_11 + b_1_23·b_3_10 + b_1_12·b_1_2·b_3_10 + b_1_13·b_3_10 + b_1_0·b_1_32·b_3_11 + b_1_0·b_1_12·b_3_11 + b_1_03·b_3_11 + b_1_05·b_1_1 + b_1_06
- b_3_112 + b_1_3·b_5_22 + b_1_33·b_3_11 + b_1_13·b_3_11 + b_1_13·b_3_10 + b_1_0·b_5_24
+ b_1_0·b_1_35 + b_1_0·b_1_12·b_3_11 + b_1_02·b_1_1·b_3_11 + b_1_03·b_3_11 + b_1_05·b_1_1 + b_1_06 + c_4_16·b_1_12 + c_4_16·b_1_02
- b_1_0·b_1_1·b_5_22 + b_1_0·b_1_13·b_3_11 + b_1_02·b_5_23 + b_1_02·b_5_22
+ b_1_04·b_3_11 + b_1_06·b_1_1 + b_1_07 + c_4_16·b_1_0·b_1_12 + c_4_16·b_1_03
- b_1_2·b_1_3·b_5_23 + b_1_22·b_5_24 + b_1_22·b_5_23 + b_1_24·b_3_11 + b_1_24·b_3_10
+ b_1_12·b_5_22 + b_1_13·b_1_2·b_3_10 + b_1_14·b_3_10 + b_1_15·b_1_22 + b_1_0·b_1_3·b_5_23 + b_1_0·b_1_13·b_3_11 + b_1_02·b_5_22 + b_1_03·b_1_1·b_3_11 + c_4_16·b_1_1·b_1_22 + c_4_16·b_1_13 + c_4_16·b_1_02·b_1_1
- b_3_10·b_5_22 + b_1_14·b_1_2·b_3_10 + b_1_0·b_1_34·b_3_11 + c_4_16·b_1_2·b_3_11
+ c_4_16·b_1_23·b_1_3 + c_4_16·b_1_12·b_1_22
- b_3_10·b_5_24 + b_3_10·b_5_23 + b_1_2·b_7_40 + b_1_23·b_5_24 + b_1_12·b_1_2·b_5_24
+ b_1_13·b_5_22 + b_1_14·b_1_2·b_3_10 + b_1_17·b_1_2 + b_1_0·b_1_34·b_3_11 + b_1_0·b_1_37 + b_1_03·b_5_23 + b_1_03·b_5_22 + b_1_04·b_1_1·b_3_11 + b_1_05·b_3_11 + b_1_07·b_1_1 + b_1_08 + c_4_16·b_1_24 + c_4_16·b_1_12·b_1_22 + c_4_16·b_1_13·b_1_2 + c_4_16·b_1_14 + c_4_16·b_1_0·b_1_33 + c_4_16·b_1_04
- b_3_10·b_5_23 + b_1_3·b_7_40 + b_1_35·b_3_11 + b_1_38 + b_1_23·b_5_24 + b_1_25·b_3_10
+ b_1_28 + b_1_14·b_1_2·b_3_10 + b_1_15·b_3_10 + b_1_03·b_5_23 + b_1_04·b_1_1·b_3_11 + b_1_05·b_3_11 + b_1_07·b_1_1 + b_1_08 + c_4_16·b_1_3·b_3_11 + c_4_16·b_1_2·b_3_10 + c_4_16·b_1_24 + c_4_16·b_1_13·b_1_2 + c_4_16·b_1_0·b_1_33 + c_4_16·b_1_03·b_1_1 + c_4_16·b_1_04
- b_1_0·b_7_40 + b_1_0·b_1_34·b_3_11 + b_1_0·b_1_37 + b_1_03·b_5_22
+ b_1_04·b_1_1·b_3_11 + b_1_05·b_3_11 + b_1_07·b_1_1 + b_1_08 + c_4_16·b_1_0·b_3_11 + c_4_16·b_1_0·b_1_13 + c_4_16·b_1_03·b_1_1 + c_4_16·b_1_04
- b_3_11·b_5_24 + b_3_11·b_5_23 + b_3_11·b_5_22 + b_3_10·b_5_23 + b_1_1·b_7_40
+ b_1_12·b_1_2·b_5_24 + b_1_13·b_5_24 + b_1_15·b_3_11 + b_1_15·b_3_10 + b_1_16·b_1_22 + b_1_17·b_1_2 + b_1_18 + b_1_0·b_1_32·b_5_23 + b_1_0·b_1_34·b_3_11 + b_1_03·b_5_23 + c_4_16·b_1_34 + c_4_16·b_1_2·b_3_10 + c_4_16·b_1_23·b_1_3 + c_4_16·b_1_24 + c_4_16·b_1_1·b_3_11 + c_4_16·b_1_14 + c_4_16·b_1_0·b_1_33 + c_4_16·b_1_03·b_1_1 + c_4_16·b_1_04
- b_1_2·b_3_11·b_5_23 + b_1_22·b_7_40 + b_1_24·b_5_24 + b_1_1·b_3_11·b_5_22
+ b_1_14·b_5_22 + b_1_15·b_1_2·b_3_10 + b_1_16·b_3_10 + b_1_0·b_3_11·b_5_23 + b_1_0·b_3_11·b_5_22 + b_1_04·b_5_23 + b_1_04·b_5_22 + c_4_16·b_1_24·b_1_3 + c_4_16·b_1_12·b_3_11 + c_4_16·b_1_13·b_1_22 + c_4_16·b_1_15 + c_4_16·b_1_0·b_1_3·b_3_11 + c_4_16·b_1_0·b_1_34 + c_4_16·b_1_0·b_1_1·b_3_11 + c_4_16·b_1_04·b_1_1
- b_5_22·b_5_24 + b_5_22·b_5_23 + b_5_222 + b_1_22·b_1_3·b_7_40 + b_1_25·b_5_24
+ b_1_12·b_3_11·b_5_22 + b_1_12·b_1_2·b_7_40 + b_1_14·b_1_2·b_5_24 + b_1_16·b_1_2·b_3_10 + b_1_18·b_1_22 + b_1_19·b_1_2 + b_1_02·b_3_11·b_5_22 + b_1_05·b_5_22 + c_4_16·b_1_33·b_3_11 + c_4_16·b_1_36 + c_4_16·b_1_2·b_5_23 + c_4_16·b_1_23·b_3_11 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_1·b_5_24 + c_4_16·b_1_13·b_3_10 + c_4_16·b_1_0·b_5_22 + c_4_16·b_1_0·b_1_32·b_3_11 + c_4_16·b_1_02·b_1_1·b_3_11 + c_4_16·b_1_03·b_3_11 + c_4_16·b_1_06 + c_4_162·b_1_22 + c_4_162·b_1_1·b_1_2 + c_4_162·b_1_12 + c_4_162·b_1_0·b_1_3 + c_4_162·b_1_0·b_1_1
- b_5_22·b_5_24 + b_5_22·b_5_23 + b_5_222 + b_3_10·b_7_40 + b_1_23·b_7_40
+ b_1_25·b_5_24 + b_1_12·b_3_11·b_5_22 + b_1_12·b_1_2·b_7_40 + b_1_14·b_1_2·b_5_24 + b_1_17·b_3_10 + b_1_18·b_1_22 + b_1_19·b_1_2 + b_1_0·b_1_39 + b_1_02·b_3_11·b_5_22 + b_1_05·b_5_22 + c_4_16·b_1_33·b_3_11 + c_4_16·b_1_36 + c_4_16·b_1_2·b_5_24 + c_4_16·b_1_23·b_3_11 + c_4_16·b_1_23·b_3_10 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_1·b_5_24 + c_4_16·b_1_12·b_1_2·b_3_10 + c_4_16·b_1_0·b_5_22 + c_4_16·b_1_0·b_1_35 + c_4_16·b_1_02·b_1_1·b_3_11 + c_4_16·b_1_03·b_3_11 + c_4_16·b_1_06 + c_4_162·b_1_22 + c_4_162·b_1_1·b_1_2 + c_4_162·b_1_12 + c_4_162·b_1_0·b_1_3 + c_4_162·b_1_0·b_1_1
- b_5_22·b_5_24 + b_5_22·b_5_23 + b_5_222 + b_3_11·b_7_40 + b_1_37·b_3_11
+ b_1_23·b_7_40 + b_1_25·b_5_24 + b_1_27·b_3_10 + b_1_12·b_3_11·b_5_22 + b_1_12·b_1_2·b_7_40 + b_1_14·b_1_2·b_5_24 + b_1_17·b_3_11 + b_1_17·b_3_10 + b_1_18·b_1_22 + b_1_19·b_1_2 + b_1_0·b_1_3·b_3_11·b_5_23 + b_1_0·b_1_36·b_3_11 + b_1_05·b_5_23 + b_1_05·b_5_22 + b_1_06·b_1_1·b_3_11 + c_4_16·b_1_33·b_3_11 + c_4_16·b_1_23·b_3_11 + c_4_16·b_1_23·b_3_10 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_26 + c_4_16·b_1_12·b_1_2·b_3_10 + c_4_16·b_1_13·b_3_10 + c_4_16·b_1_0·b_5_22 + c_4_16·b_1_0·b_1_32·b_3_11 + c_4_16·b_1_0·b_1_12·b_3_11 + c_4_16·b_1_02·b_1_1·b_3_11 + c_4_16·b_1_03·b_3_11 + c_4_16·b_1_06 + c_4_162·b_1_32 + c_4_162·b_1_12 + c_4_162·b_1_0·b_1_3 + c_4_162·b_1_0·b_1_1
- b_5_242 + b_5_232 + b_5_222 + b_1_23·b_7_40 + b_1_25·b_5_24 + b_1_27·b_3_11
+ b_1_27·b_3_10 + b_1_14·b_1_2·b_5_24 + b_1_15·b_5_22 + b_1_17·b_3_11 + b_1_0·b_1_36·b_3_11 + b_1_05·b_5_23 + b_1_05·b_5_22 + b_1_07·b_3_11 + b_1_09·b_1_1 + c_8_49·b_1_22 + c_4_16·b_1_36 + c_4_16·b_1_23·b_3_11 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_15·b_1_2 + c_4_16·b_1_0·b_1_12·b_3_11 + c_4_16·b_1_03·b_3_11
- b_5_23·b_5_24 + b_5_232 + b_5_22·b_5_23 + b_1_32·b_3_11·b_5_23 + b_1_23·b_7_40
+ b_1_25·b_5_24 + b_1_27·b_3_11 + b_1_27·b_3_10 + b_1_12·b_3_11·b_5_22 + b_1_13·b_7_40 + b_1_14·b_1_2·b_5_24 + b_1_15·b_5_24 + b_1_15·b_5_22 + b_1_17·b_3_11 + b_1_19·b_1_2 + b_1_110 + b_1_0·b_1_3·b_3_11·b_5_23 + b_1_0·b_1_34·b_5_23 + b_1_0·b_1_36·b_3_11 + b_1_02·b_3_11·b_5_22 + b_1_05·b_5_23 + b_1_05·b_5_22 + b_1_06·b_1_1·b_3_11 + b_1_09·b_1_1 + b_1_010 + c_8_49·b_1_2·b_1_3 + c_4_16·b_1_2·b_5_24 + c_4_16·b_1_2·b_5_23 + c_4_16·b_1_23·b_3_11 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_12·b_1_2·b_3_10 + c_4_16·b_1_02·b_1_1·b_3_11 + c_4_16·b_1_05·b_1_1 + c_4_162·b_1_2·b_1_3 + c_4_162·b_1_1·b_1_2
- b_5_232 + b_1_35·b_5_23 + b_1_37·b_3_11 + b_1_23·b_7_40 + b_1_25·b_5_23
+ b_1_27·b_3_10 + b_1_17·b_3_11 + b_1_17·b_3_10 + b_1_0·b_1_39 + b_1_06·b_1_1·b_3_11 + c_8_49·b_1_32 + c_4_16·b_1_23·b_3_11 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_16 + c_4_16·b_1_0·b_1_32·b_3_11 + c_4_16·b_1_06 + c_4_162·b_1_22
- b_5_22·b_5_23 + b_1_32·b_3_11·b_5_23 + b_1_12·b_3_11·b_5_22
+ b_1_0·b_1_3·b_3_11·b_5_23 + b_1_0·b_1_36·b_3_11 + b_1_02·b_3_11·b_5_23 + b_1_02·b_3_11·b_5_22 + b_1_05·b_5_23 + b_1_09·b_1_1 + b_1_010 + c_8_49·b_1_0·b_1_3 + c_4_16·b_1_3·b_5_23 + c_4_16·b_1_1·b_5_22 + c_4_16·b_1_12·b_1_2·b_3_10 + c_4_16·b_1_13·b_3_10 + c_4_16·b_1_0·b_5_23 + c_4_16·b_1_0·b_5_22 + c_4_16·b_1_0·b_1_32·b_3_11 + c_4_16·b_1_0·b_1_12·b_3_11 + c_4_16·b_1_03·b_3_11 + c_4_16·b_1_05·b_1_1 + c_4_16·b_1_06 + c_4_162·b_1_1·b_1_2 + c_4_162·b_1_12 + c_4_162·b_1_0·b_1_3 + c_4_162·b_1_0·b_1_1
- b_5_222 + b_1_16·b_1_2·b_3_10 + b_1_0·b_1_36·b_3_11 + b_1_05·b_5_23
+ b_1_05·b_5_22 + b_1_09·b_1_1 + c_8_49·b_1_02 + c_4_16·b_1_36 + c_4_16·b_1_14·b_1_22 + c_4_16·b_1_03·b_3_11 + c_4_16·b_1_05·b_1_1 + c_4_16·b_1_06 + c_4_162·b_1_32 + c_4_162·b_1_12 + c_4_162·b_1_02
- b_5_22·b_7_40 + b_1_39·b_3_11 + b_1_24·b_1_3·b_7_40 + b_1_25·b_7_40 + b_1_29·b_3_10
+ b_1_14·b_3_11·b_5_22 + b_1_18·b_1_2·b_3_10 + b_1_110·b_1_22 + b_1_0·b_1_36·b_5_23 + b_1_0·b_1_38·b_3_11 + b_1_0·b_1_311 + b_1_04·b_3_11·b_5_23 + b_1_04·b_3_11·b_5_22 + b_1_07·b_5_23 + b_1_07·b_5_22 + b_1_08·b_1_1·b_3_11 + b_1_09·b_3_11 + b_1_012 + c_8_49·b_1_04 + c_4_16·b_3_11·b_5_22 + c_4_16·b_1_3·b_7_40 + c_4_16·b_1_38 + c_4_16·b_1_23·b_5_23 + c_4_16·b_1_25·b_3_10 + c_4_16·b_1_28 + c_4_16·b_1_1·b_7_40 + c_4_16·b_1_12·b_1_2·b_5_24 + c_4_16·b_1_13·b_5_22 + c_4_16·b_1_15·b_3_11 + c_4_16·b_1_17·b_1_2 + c_4_16·b_1_0·b_1_32·b_5_23 + c_4_16·b_1_03·b_5_23 + c_4_16·b_1_03·b_5_22 + c_4_16·b_1_04·b_1_1·b_3_11 + c_4_16·b_1_05·b_3_11 + c_4_16·b_1_07·b_1_1 + c_4_16·b_1_08 + c_4_162·b_1_3·b_3_11 + c_4_162·b_1_24 + c_4_162·b_1_1·b_3_11 + c_4_162·b_1_12·b_1_22 + c_4_162·b_1_13·b_1_2 + c_4_162·b_1_14 + c_4_162·b_1_0·b_1_13 + c_4_162·b_1_03·b_1_1
- b_5_24·b_7_40 + b_5_23·b_7_40 + b_5_22·b_7_40 + b_1_39·b_3_11 + b_1_24·b_1_3·b_7_40
+ b_1_25·b_7_40 + b_1_29·b_3_10 + b_1_14·b_3_11·b_5_22 + b_1_14·b_1_2·b_7_40 + b_1_16·b_1_2·b_5_24 + b_1_17·b_5_24 + b_1_18·b_1_2·b_3_10 + b_1_19·b_3_11 + b_1_19·b_3_10 + b_1_111·b_1_2 + b_1_0·b_1_36·b_5_23 + b_1_04·b_3_11·b_5_22 + b_1_08·b_1_1·b_3_11 + b_1_011·b_1_1 + c_8_49·b_1_2·b_3_10 + c_8_49·b_1_23·b_1_3 + c_8_49·b_1_24 + c_8_49·b_1_12·b_1_22 + c_8_49·b_1_0·b_1_13 + c_8_49·b_1_03·b_1_1 + c_4_16·b_1_3·b_7_40 + c_4_16·b_1_35·b_3_11 + c_4_16·b_1_23·b_5_23 + c_4_16·b_1_27·b_1_3 + c_4_16·b_1_28 + c_4_16·b_1_1·b_7_40 + c_4_16·b_1_13·b_5_24 + c_4_16·b_1_13·b_5_22 + c_4_16·b_1_15·b_3_11 + c_4_16·b_1_15·b_3_10 + c_4_16·b_1_17·b_1_2 + c_4_16·b_1_0·b_1_37 + c_4_16·b_1_03·b_5_23 + c_4_16·b_1_03·b_5_22 + c_4_162·b_1_3·b_3_11 + c_4_162·b_1_34 + c_4_162·b_1_1·b_3_11 + c_4_162·b_1_12·b_1_22 + c_4_162·b_1_04
- b_5_23·b_7_40 + b_5_22·b_7_40 + b_1_34·b_3_11·b_5_23 + b_1_37·b_5_23 + b_1_39·b_3_11
+ b_1_24·b_1_3·b_7_40 + b_1_27·b_5_23 + b_1_29·b_3_11 + b_1_19·b_3_11 + b_1_0·b_1_33·b_3_11·b_5_23 + b_1_0·b_1_311 + b_1_07·b_5_22 + b_1_09·b_3_11 + b_1_011·b_1_1 + c_8_49·b_1_2·b_3_11 + c_8_49·b_1_23·b_1_3 + c_8_49·b_1_24 + c_8_49·b_1_1·b_3_10 + c_8_49·b_1_0·b_1_33 + c_8_49·b_1_03·b_1_1 + c_4_16·b_3_11·b_5_23 + c_4_16·b_3_11·b_5_22 + c_4_16·b_1_3·b_7_40 + c_4_16·b_1_38 + c_4_16·b_1_2·b_7_40 + c_4_16·b_1_23·b_5_23 + c_4_16·b_1_25·b_3_10 + c_4_16·b_1_1·b_7_40 + c_4_16·b_1_12·b_1_2·b_5_24 + c_4_16·b_1_13·b_5_24 + c_4_16·b_1_14·b_1_2·b_3_10 + c_4_16·b_1_15·b_3_11 + c_4_16·b_1_15·b_3_10 + c_4_16·b_1_0·b_1_34·b_3_11 + c_4_16·b_1_03·b_5_23 + c_4_16·b_1_03·b_5_22 + c_4_16·b_1_08 + c_4_162·b_1_3·b_3_11 + c_4_162·b_1_2·b_3_11 + c_4_162·b_1_24 + c_4_162·b_1_1·b_3_11 + c_4_162·b_1_12·b_1_22 + c_4_162·b_1_14 + c_4_162·b_1_0·b_1_13 + c_4_162·b_1_03·b_1_1
- b_7_402 + b_1_314 + b_1_214 + b_1_16·b_1_2·b_7_40 + b_1_110·b_1_2·b_3_10
+ b_1_112·b_1_22 + b_1_113·b_1_2 + b_1_114 + b_1_0·b_1_310·b_3_11 + b_1_09·b_5_23 + b_1_09·b_5_22 + b_1_013·b_1_1 + b_1_014 + c_8_49·b_1_12·b_1_2·b_3_10 + c_8_49·b_1_14·b_1_22 + c_8_49·b_1_0·b_1_12·b_3_11 + c_8_49·b_1_03·b_3_11 + c_8_49·b_1_06 + c_4_16·b_1_310 + c_4_16·b_1_23·b_7_40 + c_4_16·b_1_25·b_5_24 + c_4_16·b_1_27·b_3_11 + c_4_16·b_1_27·b_3_10 + c_4_16·b_1_210 + c_4_16·b_1_14·b_1_2·b_5_24 + c_4_16·b_1_15·b_5_22 + c_4_16·b_1_16·b_1_2·b_3_10 + c_4_16·b_1_18·b_1_22 + c_4_16·b_1_19·b_1_2 + c_4_16·b_1_05·b_5_23 + c_4_16·b_1_05·b_5_22 + c_4_16·b_1_06·b_1_1·b_3_11 + c_4_16·c_8_49·b_1_22 + c_4_162·b_1_23·b_3_11 + c_4_162·b_1_25·b_1_3 + c_4_162·b_1_26 + c_4_162·b_1_13·b_3_11 + c_4_162·b_1_13·b_3_10 + c_4_162·b_1_14·b_1_22 + c_4_162·b_1_15·b_1_2 + c_4_162·b_1_0·b_5_22 + c_4_162·b_1_03·b_3_11 + c_4_163·b_1_32 + c_4_163·b_1_12 + c_4_163·b_1_0·b_1_3 + c_4_163·b_1_0·b_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 14.
- 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:
- c_4_16, a Duflot regular element of degree 4
- c_8_49, a Duflot regular element of degree 8
- b_1_32 + b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 11].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 2
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_3_10 → 0, an element of degree 3
- b_3_11 → 0, an element of degree 3
- c_4_16 → c_1_04, an element of degree 4
- b_5_22 → 0, an element of degree 5
- b_5_23 → 0, an element of degree 5
- b_5_24 → 0, an element of degree 5
- b_7_40 → 0, an element of degree 7
- c_8_49 → c_1_18, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_3_10 → 0, an element of degree 3
- b_3_11 → c_1_23 + c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_22 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_23 → 0, an element of degree 5
- b_5_24 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_7_40 → c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_02·c_1_1·c_1_24
+ c_1_02·c_1_12·c_1_23 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
- c_8_49 → c_1_28 + c_1_12·c_1_26 + c_1_18 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25
+ c_1_02·c_1_12·c_1_24 + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_3_10 → 0, an element of degree 3
- b_3_11 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- c_4_16 → c_1_0·c_1_23 + c_1_04, an element of degree 4
- b_5_22 → c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_23 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- b_5_24 → c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_7_40 → c_1_27 + c_1_0·c_1_26 + c_1_02·c_1_25 + c_1_03·c_1_24 + c_1_04·c_1_23
+ c_1_05·c_1_22 + c_1_06·c_1_2, an element of degree 7
- c_8_49 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_02·c_1_26, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_3_11 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_22 → c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_23 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_24 → 0, an element of degree 5
- b_7_40 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_12·c_1_23
+ c_1_02·c_1_14·c_1_2 + c_1_03·c_1_24 + c_1_04·c_1_23 + c_1_05·c_1_22 + c_1_06·c_1_2, an element of degree 7
- c_8_49 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_14·c_1_23
+ c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_3_10 → 0, an element of degree 3
- b_3_11 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_22 → c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_5_23 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- b_5_24 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- b_7_40 → c_1_27 + c_1_0·c_1_26 + c_1_02·c_1_25 + c_1_03·c_1_24 + c_1_04·c_1_23
+ c_1_05·c_1_22 + c_1_06·c_1_2, an element of degree 7
- c_8_49 → c_1_12·c_1_26 + c_1_18 + c_1_02·c_1_26 + c_1_04·c_1_24, an element of degree 8
|