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David J. Green
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Cohomology of group number 852 of order 128
General information on the group
- The group has 3 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 1.
- It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3 and 4, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
t5 − t4 + t2 − t + 1 |
| (t − 1)4 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-5,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 14 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_2_4, an element of degree 2
- b_2_5, an element of degree 2
- b_3_8, an element of degree 3
- b_3_9, an element of degree 3
- b_3_10, an element of degree 3
- b_4_16, an element of degree 4
- b_5_21, an element of degree 5
- b_5_22, an element of degree 5
- b_6_32, an element of degree 6
- b_7_41, an element of degree 7
- c_8_57, a Duflot regular element of degree 8
Ring relations
There are 51 minimal relations of maximal degree 14:
- a_1_02
- a_1_0·b_1_1
- b_1_1·b_1_22 + a_1_0·b_1_22
- b_2_4·b_1_1 + a_1_0·b_1_22
- a_1_0·b_1_22 + b_2_5·a_1_0
- b_1_24 + b_2_4·b_2_5
- a_1_0·b_3_8
- b_1_24 + b_2_4·b_1_22 + a_1_0·b_3_9
- b_1_24 + b_1_1·b_3_9 + b_2_5·b_1_22
- b_1_24 + b_2_5·b_1_22 + a_1_0·b_3_10
- b_1_22·b_3_8
- b_2_4·b_3_8
- b_1_22·b_3_9 + b_2_5·b_3_9
- b_1_22·b_3_10 + b_2_5·b_3_9 + b_2_4·b_2_5·a_1_0
- b_2_5·b_3_9 + b_2_4·b_3_10 + b_2_4·b_2_5·a_1_0
- b_4_16·a_1_0
- b_3_8·b_3_9
- b_3_9·b_3_10 + b_4_16·b_1_22 + b_2_4·a_1_0·b_3_10
- b_3_92 + b_2_4·b_4_16 + b_2_4·a_1_0·b_3_10 + b_2_4·a_1_0·b_3_9
- b_3_82 + b_1_13·b_3_10 + b_4_16·b_1_12 + b_2_5·b_1_1·b_3_8 + b_2_53 + b_2_42·b_2_5
- a_1_0·b_5_21
- b_3_102 + b_1_1·b_5_21 + b_2_5·b_4_16
- a_1_0·b_5_22 + b_2_4·a_1_0·b_3_9
- b_1_1·b_5_22 + b_2_4·a_1_0·b_3_10
- b_1_22·b_5_21
- b_1_22·b_5_22 + b_2_5·b_5_22
- b_2_5·b_5_22 + b_2_4·b_5_22 + b_2_4·b_5_21 + b_2_42·b_3_10 + b_2_42·b_3_9
+ b_2_42·b_2_5·a_1_0
- b_6_32·a_1_0 + b_2_42·b_2_5·a_1_0
- b_6_32·b_1_1 + b_4_16·b_3_8 + b_2_5·b_5_21 + b_2_42·b_2_5·a_1_0
- b_3_8·b_5_22
- b_3_10·b_5_22 + b_3_9·b_5_22 + b_3_9·b_5_21 + b_2_4·b_2_5·b_4_16 + b_2_42·b_4_16
+ b_2_42·a_1_0·b_3_9
- b_3_10·b_5_22 + b_6_32·b_1_22 + b_2_43·b_2_5
- b_3_10·b_5_22 + b_3_9·b_5_21 + b_2_4·b_6_32 + b_2_4·b_2_5·b_4_16 + b_2_42·b_4_16
+ b_2_43·b_2_5 + b_2_42·a_1_0·b_3_10
- a_1_0·b_7_41 + b_2_42·a_1_0·b_3_10
- b_3_8·b_5_21 + b_1_1·b_7_41 + b_2_5·b_1_1·b_5_21 + b_2_5·b_4_16·b_1_12
+ b_2_52·b_1_1·b_3_10 + b_2_52·b_4_16 + b_2_4·b_2_5·b_4_16
- b_6_32·b_3_9 + b_4_16·b_5_22 + b_2_43·b_3_10 + b_2_43·b_2_5·a_1_0
- b_6_32·b_3_8 + b_4_16·b_1_12·b_3_10 + b_4_162·b_1_1 + b_2_5·b_7_41
+ b_2_5·b_4_16·b_3_8 + b_2_52·b_5_21 + b_2_52·b_4_16·b_1_1 + b_2_53·b_3_10 + b_2_43·b_2_5·a_1_0
- b_1_22·b_7_41 + b_2_43·b_3_10 + b_2_43·b_2_5·a_1_0
- b_2_4·b_7_41 + b_2_4·b_4_16·b_3_10 + b_2_4·b_4_16·b_3_9 + b_2_42·b_5_21
+ b_2_43·b_3_10 + b_2_43·b_2_5·a_1_0
- b_5_222 + b_2_4·b_4_162 + b_2_43·b_4_16 + b_2_43·a_1_0·b_3_10
+ b_2_43·a_1_0·b_3_9
- b_5_21·b_5_22 + b_4_162·b_1_22 + b_2_4·b_4_162 + b_2_4·b_2_5·b_6_32
+ b_2_42·b_6_32 + b_2_42·b_2_5·b_4_16 + b_2_43·b_4_16
- b_3_8·b_7_41 + b_1_12·b_3_10·b_5_21 + b_4_16·b_1_1·b_5_21 + b_2_5·b_4_16·b_1_1·b_3_8
+ b_2_52·b_3_8·b_3_10 + b_2_52·b_6_32 + b_2_4·b_2_5·b_6_32
- b_5_21·b_5_22 + b_3_9·b_7_41 + b_2_42·b_2_5·b_4_16
- b_5_212 + b_1_13·b_7_41 + b_1_14·b_3_8·b_3_10 + b_1_15·b_5_21 + b_4_16·b_1_1·b_5_21
+ b_4_16·b_1_16 + b_4_162·b_1_12 + b_2_5·b_1_12·b_3_8·b_3_10 + b_2_5·b_1_13·b_5_21 + b_2_5·b_1_15·b_3_10 + b_2_5·b_1_15·b_3_8 + b_2_5·b_4_16·b_1_1·b_3_8 + b_2_5·b_4_16·b_1_14 + b_2_5·b_4_162 + b_2_52·b_1_1·b_5_21 + b_2_52·b_1_13·b_3_8 + b_2_52·b_4_16·b_1_12 + b_2_54·b_1_12 + b_2_4·b_4_162 + c_8_57·b_1_12
- b_6_32·b_5_22 + b_4_162·b_3_9 + b_2_42·b_4_16·b_3_9 + b_2_43·b_5_22 + b_2_43·b_5_21
+ b_2_44·b_3_10 + b_2_44·b_3_9 + b_2_44·b_2_5·a_1_0
- b_6_32·b_5_21 + b_4_16·b_7_41 + b_2_5·b_1_12·b_7_41 + b_2_5·b_1_13·b_3_8·b_3_10
+ b_2_5·b_1_14·b_5_21 + b_2_5·b_4_16·b_1_15 + b_2_52·b_1_1·b_3_8·b_3_10 + b_2_52·b_1_12·b_5_21 + b_2_52·b_1_14·b_3_10 + b_2_52·b_1_14·b_3_8 + b_2_52·b_4_16·b_3_10 + b_2_52·b_4_16·b_3_8 + b_2_52·b_4_16·b_1_13 + b_2_53·b_5_21 + b_2_53·b_1_12·b_3_8 + b_2_53·b_4_16·b_1_1 + b_2_55·b_1_1 + b_2_44·b_2_5·a_1_0 + b_2_5·c_8_57·b_1_1 + b_2_5·c_8_57·a_1_0
- b_5_22·b_7_41 + b_4_16·b_6_32·b_1_22 + b_2_4·b_4_16·b_6_32 + b_2_4·b_2_5·b_4_162
+ b_2_42·b_4_162 + b_2_43·b_6_32 + b_2_43·b_2_5·b_4_16 + b_2_44·b_4_16 + b_2_45·b_2_5
- b_6_322 + b_4_16·b_6_32·b_1_22 + b_4_162·b_1_1·b_3_10 + b_4_163
+ b_2_5·b_4_16·b_6_32 + b_2_52·b_1_1·b_7_41 + b_2_52·b_1_12·b_3_8·b_3_10 + b_2_52·b_1_13·b_5_21 + b_2_52·b_4_16·b_1_14 + b_2_52·b_4_162 + b_2_53·b_3_8·b_3_10 + b_2_53·b_1_1·b_5_21 + b_2_53·b_1_13·b_3_10 + b_2_53·b_1_13·b_3_8 + b_2_53·b_6_32 + b_2_53·b_4_16·b_1_12 + b_2_54·b_1_1·b_3_8 + b_2_54·b_4_16 + b_2_56 + b_2_4·b_2_5·b_4_162 + b_2_42·b_4_162 + b_2_42·b_2_5·b_6_32 + b_2_43·b_2_5·b_4_16 + b_2_44·a_1_0·b_3_10 + b_2_52·c_8_57 + b_2_4·b_2_5·c_8_57
- b_5_21·b_7_41 + b_1_14·b_3_10·b_5_21 + b_1_15·b_7_41 + b_1_17·b_5_21
+ b_4_16·b_1_1·b_7_41 + b_4_16·b_1_13·b_5_21 + b_4_16·b_1_15·b_3_10 + b_4_16·b_1_15·b_3_8 + b_4_162·b_1_1·b_3_8 + b_2_5·b_1_14·b_3_8·b_3_10 + b_2_5·b_1_15·b_5_21 + b_2_5·b_1_17·b_3_10 + b_2_5·b_4_16·b_1_1·b_5_21 + b_2_5·b_4_16·b_6_32 + b_2_5·b_4_162·b_1_12 + b_2_52·b_3_10·b_5_21 + b_2_52·b_1_1·b_7_41 + b_2_52·b_1_12·b_3_8·b_3_10 + b_2_52·b_1_15·b_3_10 + b_2_52·b_4_16·b_1_1·b_3_10 + b_2_52·b_4_16·b_1_14 + b_2_53·b_1_1·b_5_21 + b_2_53·b_4_16·b_1_12 + b_2_54·b_1_1·b_3_8 + b_2_54·b_1_14 + b_2_4·b_4_16·b_6_32 + b_2_44·a_1_0·b_3_10 + c_8_57·b_1_1·b_3_8 + b_2_5·c_8_57·b_1_12
- b_6_32·b_7_41 + b_4_16·b_1_1·b_3_10·b_5_21 + b_4_162·b_5_21
+ b_2_5·b_1_13·b_3_10·b_5_21 + b_2_5·b_1_14·b_7_41 + b_2_5·b_1_16·b_5_21 + b_2_5·b_4_16·b_7_41 + b_2_5·b_4_16·b_1_12·b_5_21 + b_2_5·b_4_16·b_1_14·b_3_10 + b_2_5·b_4_16·b_1_14·b_3_8 + b_2_52·b_1_13·b_3_8·b_3_10 + b_2_52·b_1_14·b_5_21 + b_2_52·b_1_16·b_3_10 + b_2_52·b_6_32·b_3_10 + b_2_52·b_4_16·b_5_21 + b_2_52·b_4_162·b_1_1 + b_2_53·b_7_41 + b_2_53·b_1_1·b_3_8·b_3_10 + b_2_53·b_1_14·b_3_10 + b_2_53·b_4_16·b_3_10 + b_2_53·b_4_16·b_1_13 + b_2_54·b_5_21 + b_2_54·b_4_16·b_1_1 + b_2_55·b_3_8 + b_2_55·b_1_13 + b_2_42·b_4_16·b_5_21 + b_2_45·b_3_10 + b_2_5·c_8_57·b_3_8 + b_2_52·c_8_57·b_1_1 + b_2_4·b_2_5·c_8_57·a_1_0
- b_7_412 + b_1_14·b_3_10·b_7_41 + b_1_16·b_3_10·b_5_21 + b_1_17·b_7_41
+ b_4_16·b_1_12·b_3_10·b_5_21 + b_4_16·b_1_13·b_7_41 + b_4_16·b_1_14·b_3_8·b_3_10 + b_4_16·b_1_15·b_5_21 + b_4_16·b_1_17·b_3_10 + b_4_162·b_1_1·b_5_21 + b_4_162·b_1_13·b_3_10 + b_4_162·b_1_16 + b_4_163·b_1_12 + b_2_5·b_1_16·b_3_8·b_3_10 + b_2_5·b_1_17·b_5_21 + b_2_5·b_4_16·b_1_1·b_7_41 + b_2_5·b_4_16·b_1_15·b_3_10 + b_2_5·b_4_16·b_1_15·b_3_8 + b_2_5·b_4_16·b_1_18 + b_2_5·b_4_162·b_1_1·b_3_10 + b_2_5·b_4_162·b_1_14 + b_2_5·b_4_163 + b_2_52·b_1_12·b_3_10·b_5_21 + b_2_52·b_4_16·b_1_1·b_5_21 + b_2_52·b_4_16·b_1_13·b_3_10 + b_2_52·b_4_16·b_1_16 + b_2_52·b_4_16·b_6_32 + b_2_52·b_4_162·b_1_12 + b_2_53·b_1_13·b_5_21 + b_2_53·b_4_16·b_1_1·b_3_10 + b_2_53·b_4_16·b_1_14 + b_2_53·b_4_162 + b_2_54·b_3_8·b_3_10 + b_2_54·b_1_1·b_5_21 + b_2_54·b_1_13·b_3_8 + b_2_54·b_6_32 + b_2_54·b_4_16·b_1_12 + b_2_55·b_1_14 + b_2_57 + b_2_4·b_4_163 + b_2_4·b_2_5·b_4_16·b_6_32 + b_2_43·b_4_162 + b_2_43·b_2_5·b_6_32 + b_2_44·b_2_5·b_4_16 + b_2_46·b_2_5 + c_8_57·b_1_13·b_3_10 + b_4_16·c_8_57·b_1_12 + b_2_5·c_8_57·b_1_1·b_3_8 + b_2_52·c_8_57·b_1_12 + b_2_53·c_8_57 + b_2_42·b_2_5·c_8_57
Data used for Benson′s test
- Benson′s completion test succeeded in degree 17.
- However, the last relation was already found in degree 14 and the last generator in degree 8.
- The following is a filter regular homogeneous system of parameters:
- c_8_57, a Duflot regular element of degree 8
- b_1_1·b_3_10 + b_1_14 + b_4_16 + b_2_52 + b_2_4·b_2_5 + b_2_42, an element of degree 4
- b_1_2·b_5_22 + b_1_13·b_3_10 + b_4_16·b_1_12 + b_2_5·b_1_2·b_3_10 + b_2_5·b_1_2·b_3_8
+ b_2_5·b_1_1·b_3_10 + b_2_5·b_4_16 + b_2_52·b_1_12 + b_2_4·b_1_2·b_3_10 + b_2_4·b_1_2·b_3_9 + b_2_4·b_4_16, an element of degree 6
- b_1_1·b_1_2·b_5_21 + b_2_5·b_1_1·b_1_2·b_3_10 + b_2_5·b_1_12·b_3_10
+ b_2_5·b_4_16·b_1_1 + b_2_53·b_1_2 + b_2_4·b_5_22 + b_2_4·b_5_21 + b_2_4·b_4_16·b_1_2 + b_2_42·b_3_9 + b_2_43·b_1_2, an element of degree 7
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 14, 21].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
- We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 3 elements of degree 4.
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_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_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_3_8 → 0, an element of degree 3
- b_3_9 → 0, an element of degree 3
- b_3_10 → 0, an element of degree 3
- b_4_16 → 0, an element of degree 4
- b_5_21 → 0, an element of degree 5
- b_5_22 → 0, an element of degree 5
- b_6_32 → 0, an element of degree 6
- b_7_41 → 0, an element of degree 7
- c_8_57 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_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_2_4 → c_1_12, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_3_8 → 0, an element of degree 3
- b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_3_10 → 0, an element of degree 3
- b_4_16 → c_1_24 + c_1_12·c_1_22, an element of degree 4
- b_5_21 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
- b_5_22 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
- b_6_32 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_14·c_1_22, an element of degree 6
- b_7_41 → c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_15·c_1_22, an element of degree 7
- c_8_57 → c_1_28 + c_1_14·c_1_24 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
+ c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_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_2_4 → c_1_22, an element of degree 2
- b_2_5 → c_1_22, an element of degree 2
- b_3_8 → 0, an element of degree 3
- b_3_9 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_3_10 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_4_16 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_21 → 0, an element of degree 5
- b_5_22 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
- b_6_32 → c_1_26 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_15·c_1_2 + c_1_16, an element of degree 6
- b_7_41 → c_1_1·c_1_26 + c_1_12·c_1_25, an element of degree 7
- c_8_57 → c_1_12·c_1_26 + c_1_18 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22
+ c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_3_8 → c_1_23 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_12·c_1_2, an element of degree 3
- b_3_9 → 0, an element of degree 3
- b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_2·c_1_3 + c_1_02·c_1_1, an element of degree 3
- b_4_16 → c_1_34 + c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_02·c_1_12, an element of degree 4
- b_5_21 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_12·c_1_2·c_1_32 + c_1_02·c_1_1·c_1_22
+ c_1_02·c_1_12·c_1_2 + c_1_04·c_1_1, an element of degree 5
- b_5_22 → 0, an element of degree 5
- b_6_32 → c_1_36 + c_1_24·c_1_32 + c_1_1·c_1_35 + c_1_1·c_1_2·c_1_34
+ c_1_1·c_1_22·c_1_33 + c_1_1·c_1_23·c_1_32 + c_1_12·c_1_34 + c_1_12·c_1_22·c_1_32 + c_1_13·c_1_33 + c_1_13·c_1_2·c_1_32 + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_32 + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_3 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2, an element of degree 6
- b_7_41 → c_1_2·c_1_36 + c_1_26·c_1_3 + c_1_1·c_1_2·c_1_35 + c_1_1·c_1_23·c_1_33
+ c_1_1·c_1_25·c_1_3 + c_1_12·c_1_23·c_1_32 + c_1_12·c_1_24·c_1_3 + c_1_13·c_1_2·c_1_33 + c_1_13·c_1_23·c_1_3 + c_1_14·c_1_2·c_1_32 + c_1_02·c_1_25 + c_1_02·c_1_1·c_1_22·c_1_32 + c_1_02·c_1_12·c_1_2·c_1_32 + c_1_02·c_1_12·c_1_22·c_1_3 + c_1_02·c_1_12·c_1_23 + c_1_02·c_1_13·c_1_2·c_1_3 + c_1_02·c_1_13·c_1_22 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_23 + c_1_04·c_1_1·c_1_32 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_3, an element of degree 7
- c_8_57 → c_1_38 + c_1_22·c_1_36 + c_1_26·c_1_32 + c_1_27·c_1_3 + c_1_28
+ c_1_1·c_1_22·c_1_35 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_26·c_1_3 + c_1_1·c_1_27 + c_1_12·c_1_23·c_1_33 + c_1_13·c_1_25 + c_1_14·c_1_22·c_1_32 + c_1_14·c_1_23·c_1_3 + c_1_15·c_1_2·c_1_32 + c_1_16·c_1_32 + c_1_16·c_1_2·c_1_3 + c_1_16·c_1_22 + c_1_02·c_1_22·c_1_34 + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_26 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_2·c_1_32 + c_1_02·c_1_13·c_1_22·c_1_3 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_32 + c_1_02·c_1_14·c_1_2·c_1_3 + c_1_02·c_1_15·c_1_3 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_34 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_13·c_1_3 + c_1_04·c_1_13·c_1_2 + c_1_06·c_1_12 + c_1_08, an element of degree 8
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