Simon King
David J. Green
Cohomology
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Cohomology of group number 334 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 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t5 − 2·t4 + 2·t3 − t2 − 1) |
| (t + 1) · (t − 1)4 · (t2 + 1)2 |
- 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 12 minimal generators of maximal degree 7:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_1_2, an element of degree 1
- a_2_4, a nilpotent element of degree 2
- b_2_5, an element of degree 2
- a_3_7, a nilpotent element of degree 3
- b_4_7, an element of degree 4
- b_4_9, an element of degree 4
- c_4_10, a Duflot regular element of degree 4
- c_4_11, a Duflot regular element of degree 4
- b_5_17, an element of degree 5
- b_7_31, an element of degree 7
Ring relations
There are 37 minimal relations of maximal degree 14:
- a_1_0·a_1_1
- a_1_0·b_1_2
- a_1_13 + a_1_03
- a_2_4·b_1_2 + a_1_12·b_1_2
- a_1_12·b_1_2 + a_2_4·a_1_1
- b_2_5·a_1_1 + a_1_12·b_1_2
- b_1_2·a_3_7
- a_1_1·a_3_7
- a_1_0·a_3_7 + b_2_5·a_1_02 + a_2_42
- b_4_7·a_1_0 + a_2_4·b_2_5·a_1_0 + a_2_42·a_1_0
- b_4_9·a_1_1 + b_4_7·a_1_1
- b_4_9·a_1_0 + a_2_4·a_3_7 + a_2_4·b_2_5·a_1_0 + a_2_42·a_1_0
- a_3_72 + a_2_42·b_2_5 + a_2_4·b_2_5·a_1_02 + a_2_43 + c_4_10·a_1_02
- b_4_7·a_1_12
- a_2_4·b_4_7 + a_2_42·b_2_5 + a_2_4·b_2_5·a_1_02 + a_2_43
- a_2_4·b_4_9 + a_3_72 + b_2_52·a_1_02 + a_2_4·b_2_5·a_1_02 + a_2_43
- b_1_2·b_5_17 + b_2_5·b_4_7 + b_4_7·a_1_1·b_1_2 + a_2_4·b_2_52 + b_2_52·a_1_02
+ a_2_42·b_2_5 + c_4_11·b_1_22 + c_4_10·b_1_22 + c_4_10·a_1_1·b_1_2
- a_1_1·b_5_17 + c_4_11·a_1_1·b_1_2 + c_4_10·a_1_1·b_1_2 + c_4_10·a_1_12
- a_1_0·b_5_17 + a_3_72 + a_2_42·b_2_5 + a_2_43
- b_4_7·a_3_7 + a_2_4·b_2_5·a_3_7 + a_2_42·b_2_5·a_1_0 + c_4_10·a_1_03
- b_4_9·a_3_7 + a_2_4·b_2_52·a_1_0 + a_2_42·b_2_5·a_1_0 + a_2_4·c_4_10·a_1_0
- a_2_4·b_5_17 + a_2_4·c_4_11·a_1_1 + a_2_4·c_4_10·a_1_0
- b_4_72 + a_2_42·b_2_52 + c_4_11·b_1_24
- b_4_92 + b_2_52·b_4_7 + a_2_4·b_2_53 + a_2_42·b_2_52 + a_2_43·b_2_5
+ c_4_11·b_1_24 + b_2_5·c_4_10·a_1_02 + a_2_42·c_4_10 + a_2_4·c_4_10·a_1_02
- a_3_7·b_5_17 + a_2_43·b_2_5 + b_2_5·c_4_10·a_1_02 + a_2_42·c_4_10
- b_1_2·b_7_31 + b_4_7·b_1_24 + b_4_7·b_4_9 + b_2_5·b_4_9·b_1_22
+ b_2_5·b_4_7·b_1_22 + b_2_52·b_4_7 + b_4_7·a_1_1·b_1_23 + a_2_4·b_2_53 + a_2_4·b_2_52·a_1_02 + c_4_11·b_1_24 + c_4_10·b_1_24 + b_2_5·c_4_11·b_1_22 + b_2_5·c_4_10·b_1_22 + c_4_11·a_1_1·b_1_23 + b_2_5·c_4_10·a_1_02 + a_2_4·c_4_10·a_1_02
- a_1_1·b_7_31 + b_4_7·a_1_1·b_1_23 + c_4_10·a_1_1·b_1_23
- a_1_0·b_7_31 + b_2_53·a_1_02 + a_2_42·b_2_52 + a_2_43·b_2_5 + a_2_42·c_4_10
+ a_2_4·c_4_11·a_1_02 + a_2_4·c_4_10·a_1_02
- b_4_7·b_5_17 + b_4_7·c_4_11·b_1_2 + b_4_7·c_4_10·b_1_2 + b_2_5·c_4_11·b_1_23
+ c_4_11·a_1_1·b_1_24 + b_4_7·c_4_10·a_1_1 + a_2_4·b_2_5·c_4_10·a_1_0 + a_2_42·c_4_10·a_1_0
- b_4_9·b_5_17 + b_2_5·b_7_31 + b_2_5·b_4_7·b_1_23 + b_2_52·b_5_17
+ b_2_52·b_4_9·b_1_2 + b_2_52·b_4_7·b_1_2 + b_2_53·a_3_7 + a_2_4·b_2_52·a_3_7 + b_4_9·c_4_11·b_1_2 + b_4_9·c_4_10·b_1_2 + b_2_5·c_4_11·b_1_23 + b_2_5·c_4_10·b_1_23 + c_4_11·a_1_1·b_1_24 + b_4_7·c_4_10·a_1_1 + b_2_5·c_4_10·a_3_7 + a_2_4·c_4_10·a_3_7 + a_2_4·b_2_5·c_4_11·a_1_0 + a_2_42·c_4_10·a_1_0
- a_2_4·b_7_31 + a_2_4·b_2_52·a_3_7 + a_2_4·c_4_10·a_3_7 + a_2_4·b_2_5·c_4_10·a_1_0
+ a_2_42·c_4_11·a_1_0 + a_2_42·c_4_10·a_1_0
- b_5_172 + b_2_52·c_4_11·b_1_22 + c_4_112·b_1_22 + c_4_102·b_1_22
+ c_4_102·a_1_12 + c_4_102·a_1_02
- a_3_7·b_7_31 + a_2_42·b_2_53 + a_2_4·b_2_53·a_1_02 + a_2_43·b_2_52
+ a_2_4·b_2_5·c_4_11·a_1_02 + a_2_4·b_2_5·c_4_10·a_1_02 + a_2_43·c_4_11 + c_4_102·a_1_02
- b_4_7·b_7_31 + b_2_5·b_4_7·b_4_9·b_1_2 + a_2_4·b_2_53·a_3_7 + a_2_42·b_2_53·a_1_0
+ c_4_11·b_1_27 + b_4_9·c_4_11·b_1_23 + b_4_7·c_4_11·b_1_23 + b_4_7·c_4_10·b_1_23 + b_2_5·c_4_11·b_1_25 + b_2_5·b_4_7·c_4_11·b_1_2 + b_2_5·b_4_7·c_4_10·b_1_2 + b_2_52·c_4_11·b_1_23 + c_4_11·a_1_1·b_1_26 + b_4_7·c_4_11·a_1_1·b_1_22 + a_2_4·b_2_5·c_4_10·a_3_7 + a_2_4·b_2_52·c_4_10·a_1_0 + a_2_42·b_2_5·c_4_11·a_1_0 + a_2_42·b_2_5·c_4_10·a_1_0 + c_4_102·a_1_03
- b_4_9·b_7_31 + b_4_7·b_4_9·b_1_23 + b_2_5·b_4_7·b_4_9·b_1_2 + b_2_52·b_7_31
+ b_2_52·b_4_7·b_1_23 + b_2_53·b_5_17 + b_2_53·b_4_9·b_1_2 + b_2_54·a_3_7 + a_2_4·b_2_53·a_3_7 + a_2_4·b_2_54·a_1_0 + b_4_9·c_4_11·b_1_23 + b_4_9·c_4_10·b_1_23 + b_4_7·c_4_11·b_1_23 + b_2_5·c_4_11·b_1_25 + b_2_5·b_4_9·c_4_11·b_1_2 + b_2_5·b_4_9·c_4_10·b_1_2 + b_2_52·c_4_10·b_1_23 + c_4_11·a_1_1·b_1_26 + b_4_7·c_4_11·a_1_1·b_1_22 + b_2_52·c_4_10·a_3_7 + a_2_4·b_2_5·c_4_10·a_3_7 + a_2_4·b_2_52·c_4_11·a_1_0 + a_2_42·b_2_5·c_4_11·a_1_0 + a_2_4·c_4_102·a_1_0 + c_4_10·c_4_11·a_1_03 + c_4_102·a_1_03
- b_5_17·b_7_31 + b_2_52·b_4_7·b_4_9 + b_2_55·a_1_02 + a_2_42·b_2_54
+ a_2_4·b_2_54·a_1_02 + a_2_43·b_2_53 + b_4_7·c_4_11·b_1_24 + b_4_7·c_4_10·b_1_24 + b_4_7·b_4_9·c_4_11 + b_4_7·b_4_9·c_4_10 + b_2_5·c_4_11·b_1_26 + b_2_5·b_4_9·c_4_10·b_1_22 + b_2_52·c_4_11·b_1_24 + b_2_53·c_4_11·b_1_22 + c_4_11·a_1_1·b_1_27 + b_4_7·c_4_11·a_1_1·b_1_23 + b_4_7·c_4_10·a_1_1·b_1_23 + b_2_53·c_4_11·a_1_02 + b_2_53·c_4_10·a_1_02 + a_2_42·b_2_52·c_4_11 + a_2_4·b_2_52·c_4_11·a_1_02 + a_2_4·b_2_52·c_4_10·a_1_02 + c_4_112·b_1_24 + c_4_102·b_1_24 + b_2_5·c_4_112·b_1_22 + b_2_5·c_4_102·b_1_22 + c_4_112·a_1_1·b_1_23 + c_4_10·c_4_11·a_1_1·b_1_23 + c_4_102·a_1_1·b_1_23 + b_2_5·c_4_10·c_4_11·a_1_02 + b_2_5·c_4_102·a_1_02 + a_2_42·c_4_102
- b_7_312 + b_2_54·b_4_7·b_1_22 + a_2_42·b_2_55 + a_2_4·b_2_55·a_1_02
+ a_2_43·b_2_54 + c_4_11·b_1_210 + b_2_52·b_4_7·c_4_11·b_1_22 + b_2_54·c_4_11·b_1_22 + b_2_54·c_4_10·a_1_02 + c_4_102·b_1_26 + b_2_52·c_4_112·b_1_22 + b_2_52·c_4_102·b_1_22 + b_2_52·c_4_102·a_1_02 + a_2_42·b_2_5·c_4_102 + a_2_4·b_2_5·c_4_102·a_1_02 + a_2_43·c_4_102 + c_4_103·a_1_02
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_10, a Duflot regular element of degree 4
- c_4_11, a Duflot regular element of degree 4
- b_1_22 + b_2_5, an element of degree 2
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 6, 8].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- a_3_7 → 0, an element of degree 3
- b_4_7 → 0, an element of degree 4
- b_4_9 → 0, an element of degree 4
- c_4_10 → c_1_14, an element of degree 4
- c_4_11 → c_1_04, an element of degree 4
- b_5_17 → 0, an element of degree 5
- b_7_31 → 0, an element of degree 7
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- a_2_4 → 0, an element of degree 2
- b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- a_3_7 → 0, an element of degree 3
- b_4_7 → c_1_02·c_1_22, an element of degree 4
- b_4_9 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_22, an element of degree 4
- c_4_10 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
+ c_1_12·c_1_22 + c_1_14, an element of degree 4
- c_4_11 → c_1_04, an element of degree 4
- b_5_17 → c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_2·c_1_32
+ c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_2, an element of degree 5
- b_7_31 → c_1_1·c_1_22·c_1_34 + c_1_1·c_1_25·c_1_3 + c_1_12·c_1_2·c_1_34
+ c_1_12·c_1_23·c_1_32 + c_1_12·c_1_25 + c_1_14·c_1_2·c_1_32 + c_1_14·c_1_22·c_1_3 + c_1_14·c_1_23 + c_1_0·c_1_22·c_1_34 + c_1_0·c_1_24·c_1_32 + c_1_02·c_1_2·c_1_34 + c_1_02·c_1_23·c_1_32 + c_1_02·c_1_25 + c_1_03·c_1_22·c_1_32 + c_1_03·c_1_23·c_1_3 + c_1_04·c_1_2·c_1_32 + c_1_04·c_1_22·c_1_3, an element of degree 7
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