Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 1371 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 3.
- It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t5 − t4 + 3·t3 + t + 1 |
| (t + 1) · (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-5,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 13 minimal generators of maximal degree 5:
- 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_1_3, an element of degree 1
- c_2_7, a Duflot regular element of degree 2
- a_3_6, a nilpotent element of degree 3
- b_3_10, an element of degree 3
- b_3_11, an element of degree 3
- b_3_13, an element of degree 3
- b_3_14, an element of degree 3
- c_4_24, a Duflot regular element of degree 4
- c_4_25, a Duflot regular element of degree 4
- b_5_40, an element of degree 5
Ring relations
There are 36 minimal relations of maximal degree 10:
- a_1_0·b_1_2 + a_1_02
- b_1_1·b_1_2 + b_1_12 + a_1_02
- b_1_2·b_1_3 + b_1_12 + a_1_0·b_1_1
- a_1_03
- a_1_0·b_1_1·b_1_3 + a_1_02·b_1_1
- b_1_2·a_3_6 + a_1_0·a_3_6
- b_1_3·b_3_11 + b_1_3·b_3_10 + b_1_2·b_3_10 + b_1_14 + a_1_0·b_3_10 + c_2_7·b_1_22
+ c_2_7·b_1_1·b_1_3 + c_2_7·b_1_12 + c_2_7·a_1_0·b_1_3 + c_2_7·a_1_0·b_1_1
- a_1_0·b_3_11 + a_1_0·b_3_10 + c_2_7·a_1_0·b_1_1 + c_2_7·a_1_02
- b_1_2·b_3_10 + b_1_1·b_3_11 + b_1_1·b_3_10 + b_1_14 + c_2_7·b_1_22 + c_2_7·a_1_0·b_1_1
+ c_2_7·a_1_02
- b_1_1·a_3_6 + a_1_0·b_3_13 + c_2_7·a_1_02
- b_1_2·b_3_13 + b_1_1·b_3_13 + b_1_2·a_3_6 + c_2_7·b_1_22 + c_2_7·b_1_12
+ c_2_7·a_1_02
- b_1_3·b_3_14 + b_1_2·b_3_13 + b_1_3·a_3_6 + b_1_2·a_3_6 + b_1_1·a_3_6 + c_2_7·b_1_32
+ c_2_7·b_1_22 + c_2_7·b_1_12 + c_2_7·a_1_0·b_1_1
- a_1_0·b_3_14 + c_2_7·a_1_0·b_1_3 + c_2_7·a_1_02
- b_1_2·b_3_13 + b_1_1·b_3_14 + b_1_1·a_3_6 + c_2_7·b_1_22 + c_2_7·b_1_1·b_1_3
+ c_2_7·b_1_12 + c_2_7·a_1_02
- a_1_0·b_1_1·b_3_10 + a_1_02·a_3_6
- a_3_6·b_3_11 + a_3_6·b_3_10 + c_2_7·a_1_0·b_3_13 + c_2_7·a_1_0·a_3_6 + c_2_72·a_1_02
- b_3_13·b_3_14 + b_3_132 + b_3_11·b_3_13 + b_3_10·b_3_14 + b_3_10·b_3_13 + b_1_13·b_3_13
+ a_3_6·b_3_13 + a_3_6·b_3_11 + a_3_62 + c_2_7·b_1_3·b_3_13 + c_2_7·b_1_3·b_3_10 + c_2_7·b_1_2·b_3_11 + c_2_7·b_1_1·b_3_13 + c_2_7·b_1_14 + c_2_7·a_1_0·a_3_6 + c_2_72·b_1_12 + c_2_72·a_1_02
- a_3_6·b_3_14 + c_2_7·b_1_3·a_3_6 + c_2_7·a_1_0·a_3_6
- b_3_13·b_3_14 + b_3_132 + a_3_6·b_3_13 + a_3_62 + c_2_7·b_1_3·b_3_13
+ c_2_7·b_1_2·b_3_14 + c_2_7·b_1_1·b_3_13 + c_2_72·b_1_22 + c_2_72·a_1_0·b_1_1 + c_2_72·a_1_02
- b_3_142 + b_1_23·b_3_14 + b_1_23·b_3_11 + b_1_13·b_3_13 + a_3_62 + c_4_24·b_1_22
+ c_2_7·b_1_24 + c_2_7·b_1_14 + c_2_72·b_1_32
- a_3_62 + c_4_24·a_1_02 + c_2_72·a_1_02
- a_3_6·b_3_13 + c_4_24·a_1_0·b_1_1 + c_2_7·a_1_0·a_3_6 + c_2_72·a_1_0·b_1_1
- b_3_132 + b_1_16 + c_4_24·b_1_12 + c_2_72·b_1_22 + c_2_72·b_1_12
- b_3_112 + b_1_33·b_3_13 + b_1_23·b_3_11 + b_1_13·b_3_13 + b_1_33·a_3_6
+ c_4_25·b_1_32 + c_4_25·b_1_22 + c_2_7·b_1_1·b_1_33 + c_2_7·a_1_0·b_1_33 + c_2_72·b_1_22 + c_2_72·a_1_02
- b_3_102 + b_1_33·b_3_13 + b_1_13·b_3_13 + b_1_33·a_3_6 + c_4_25·b_1_32
+ c_2_7·b_1_1·b_1_33 + c_2_7·a_1_0·b_1_33 + c_4_25·a_1_02 + c_2_72·b_1_22 + c_2_72·b_1_12
- b_3_10·b_3_11 + b_3_102 + b_1_13·b_3_10 + a_1_0·b_1_32·b_3_13 + c_4_25·b_1_12
+ c_2_7·b_1_2·b_3_11 + c_2_7·b_1_1·b_3_10 + c_4_25·a_1_0·b_1_3 + c_4_25·a_1_0·b_1_1 + c_2_7·a_1_0·b_3_10 + c_2_72·b_1_22 + c_2_72·a_1_02
- b_3_11·b_3_13 + b_3_10·b_3_13 + b_3_10·b_3_11 + b_3_102 + b_1_3·b_5_40
+ b_1_1·b_1_32·b_3_10 + b_1_13·b_3_13 + a_3_6·b_3_10 + b_1_33·a_3_6 + a_1_0·b_1_32·b_3_13 + a_1_0·b_1_32·b_3_10 + c_4_25·b_1_1·b_1_3 + c_2_7·b_1_3·b_3_10 + c_2_7·b_1_34 + c_2_7·b_1_1·b_3_13 + c_2_7·b_1_1·b_3_11 + c_4_24·a_1_0·b_1_3 + c_2_7·a_1_0·b_3_10 + c_2_7·a_1_0·a_3_6 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_02
- b_3_13·b_3_14 + b_3_132 + b_3_11·b_3_14 + b_3_11·b_3_13 + b_3_112 + b_3_10·b_3_13
+ b_3_10·b_3_11 + b_3_102 + b_1_33·b_3_13 + b_1_2·b_5_40 + b_1_23·b_3_14 + b_1_13·b_3_10 + b_1_16 + a_3_6·b_3_13 + a_3_6·b_3_10 + b_1_33·a_3_6 + a_1_0·b_1_32·b_3_13 + c_4_25·b_1_32 + c_2_7·b_1_3·b_3_13 + c_2_7·b_1_3·b_3_10 + c_2_7·b_1_2·b_3_11 + c_2_7·b_1_24 + c_2_7·b_1_1·b_3_11 + c_2_7·b_1_1·b_1_33 + c_4_25·a_1_0·b_1_3 + c_4_25·a_1_0·b_1_1 + c_2_7·a_1_0·b_1_33 + c_2_7·a_1_0·a_3_6 + c_2_72·b_1_22 + c_2_72·b_1_1·b_1_3 + c_2_72·b_1_12 + c_2_72·a_1_0·b_1_3 + c_2_72·a_1_02
- a_1_0·b_5_40 + a_3_62 + c_4_25·a_1_0·b_1_1 + c_2_7·a_1_0·b_3_10 + c_2_7·a_1_0·b_1_33
+ c_2_72·a_1_02
- b_3_11·b_3_13 + b_3_10·b_3_13 + b_3_102 + b_1_33·b_3_13 + b_1_1·b_5_40 + a_3_6·b_3_13
+ a_3_6·b_3_11 + a_3_6·b_3_10 + b_1_33·a_3_6 + a_1_0·b_1_32·b_3_13 + c_4_25·b_1_32 + c_2_7·b_1_2·b_3_11 + c_2_7·b_1_1·b_3_13 + c_2_7·b_1_1·b_3_11 + c_4_25·a_1_0·b_1_1 + c_2_7·a_1_0·b_1_33 + c_2_72·b_1_12 + c_2_72·a_1_0·b_1_1 + c_2_72·a_1_02
- b_3_13·b_5_40 + b_3_11·b_5_40 + b_1_25·b_3_11 + b_1_13·b_5_40 + b_1_15·b_3_10
+ b_1_18 + b_1_32·a_3_6·b_3_10 + a_1_0·b_1_3·b_3_10·b_3_13 + c_4_25·b_1_2·b_3_14 + c_4_25·b_1_2·b_3_11 + c_4_25·b_1_24 + c_4_25·b_1_1·b_3_13 + c_4_25·b_1_1·b_3_11 + c_4_25·b_1_1·b_1_33 + c_4_24·b_1_1·b_3_11 + c_4_24·b_1_1·b_3_10 + c_4_24·b_1_14 + c_2_7·b_3_10·b_3_13 + c_2_7·b_1_33·b_3_10 + c_2_7·b_1_23·b_3_14 + c_2_7·b_1_23·b_3_11 + c_2_7·b_1_13·b_3_10 + c_4_25·b_1_3·a_3_6 + c_4_25·a_1_0·b_3_10 + c_4_25·a_1_0·b_1_33 + c_4_24·a_1_0·b_3_13 + c_4_24·a_1_0·b_3_10 + c_2_7·b_1_33·a_3_6 + c_2_7·a_1_0·b_1_32·b_3_13 + c_2_7·a_1_0·b_1_32·b_3_10 + c_4_25·a_1_0·a_3_6 + c_2_7·c_4_25·b_1_32 + c_2_7·c_4_24·b_1_12 + c_2_72·b_1_1·b_3_13 + c_2_72·b_1_1·b_3_11 + c_2_7·c_4_25·a_1_0·b_1_3 + c_2_7·c_4_25·a_1_0·b_1_1 + c_2_72·a_1_0·b_3_13 + c_2_7·c_4_25·a_1_02 + c_2_7·c_4_24·a_1_02 + c_2_72·a_1_0·a_3_6 + c_2_73·b_1_22 + c_2_73·b_1_12 + c_2_73·a_1_02
- a_3_6·b_5_40 + a_1_0·b_1_3·b_3_10·b_3_13 + c_4_25·a_1_0·b_3_13 + c_2_7·a_3_6·b_3_10
+ c_2_7·b_1_33·a_3_6 + c_4_24·a_1_0·a_3_6 + c_2_7·c_4_25·a_1_02
- b_3_14·b_5_40 + b_3_13·b_5_40 + b_1_25·b_3_14 + b_1_15·b_3_13 + c_4_25·b_1_2·b_3_14
+ c_4_25·b_1_24 + c_4_25·b_1_1·b_3_13 + c_4_25·b_1_14 + c_4_24·b_1_2·b_3_11 + c_4_24·b_1_24 + c_2_7·b_3_10·b_3_13 + c_2_7·b_1_33·b_3_13 + c_2_7·b_1_23·b_3_14 + c_2_7·b_1_23·b_3_11 + c_2_7·b_1_26 + c_2_7·b_1_1·b_1_32·b_3_10 + c_2_7·b_1_13·b_3_10 + c_2_7·b_1_16 + c_4_24·a_1_0·b_3_13 + c_2_7·a_1_0·b_1_32·b_3_13 + c_2_7·a_1_0·b_1_32·b_3_10 + c_2_7·c_4_25·b_1_22 + c_2_7·c_4_25·b_1_1·b_1_3 + c_2_7·c_4_25·b_1_12 + c_2_7·c_4_24·b_1_22 + c_2_72·b_1_3·b_3_10 + c_2_72·b_1_34 + c_2_72·b_1_2·b_3_11 + c_2_72·b_1_24 + c_2_72·b_1_1·b_3_11 + c_2_72·b_1_1·b_3_10 + c_2_72·b_1_14 + c_2_7·c_4_25·a_1_0·b_1_3 + c_2_7·c_4_24·a_1_0·b_1_3 + c_2_7·c_4_25·a_1_02 + c_2_73·b_1_12 + c_2_73·a_1_0·b_1_3
- b_3_13·b_5_40 + b_1_15·b_3_10 + a_1_0·b_1_3·b_3_10·b_3_13 + c_4_24·b_1_1·b_3_11
+ c_4_24·b_1_1·b_3_10 + c_4_24·b_1_14 + c_2_7·b_3_10·b_3_13 + c_2_7·b_1_33·b_3_13 + c_2_7·b_1_2·b_5_40 + c_4_25·a_1_0·b_3_13 + c_4_24·a_1_0·b_3_13 + c_4_25·a_1_0·a_3_6 + c_2_7·c_4_24·b_1_12 + c_2_72·b_1_1·b_3_13 + c_2_72·b_1_14 + c_2_7·c_4_24·a_1_0·b_1_1 + c_2_72·a_1_0·b_3_13 + c_2_7·c_4_25·a_1_02 + c_2_7·c_4_24·a_1_02 + c_2_72·a_1_0·a_3_6 + c_2_73·b_1_22 + c_2_73·a_1_02
- b_3_13·b_5_40 + b_3_10·b_5_40 + b_1_15·b_3_10 + b_1_32·a_3_6·b_3_10
+ a_1_0·b_1_3·b_3_10·b_3_13 + c_4_25·b_1_1·b_3_13 + c_4_25·b_1_1·b_3_11 + c_4_25·b_1_1·b_1_33 + c_4_24·b_1_1·b_3_11 + c_4_24·b_1_1·b_3_10 + c_4_24·b_1_14 + c_2_7·b_3_10·b_3_13 + c_2_7·b_1_33·b_3_10 + c_2_7·b_1_1·b_5_40 + c_4_25·b_1_3·a_3_6 + c_4_25·a_1_0·b_3_10 + c_4_25·a_1_0·b_1_33 + c_4_24·a_1_0·b_3_13 + c_4_24·a_1_0·b_3_10 + c_2_7·b_1_33·a_3_6 + c_4_25·a_1_0·a_3_6 + c_2_7·c_4_25·b_1_32 + c_2_7·c_4_24·b_1_12 + c_2_72·b_1_1·b_3_13 + c_2_72·b_1_1·b_3_10 + c_2_72·b_1_14 + c_2_72·a_1_0·b_3_13 + c_2_72·a_1_0·b_3_10 + c_2_72·a_1_0·b_1_33 + c_2_72·a_1_0·a_3_6 + c_2_73·b_1_22 + c_2_73·a_1_0·b_1_1 + c_2_73·a_1_02
- b_5_402 + b_1_25·b_5_40 + b_1_15·b_5_40 + c_4_25·b_1_23·b_3_14
+ c_4_25·b_1_23·b_3_11 + c_4_25·b_1_26 + c_4_25·b_1_13·b_3_13 + c_4_25·b_1_16 + c_4_24·b_1_23·b_3_11 + c_4_24·b_1_26 + c_2_7·b_1_25·b_3_14 + c_2_7·b_1_15·b_3_13 + c_2_7·b_1_18 + c_4_252·b_1_22 + c_4_252·b_1_12 + c_4_24·c_4_25·b_1_22 + c_2_7·c_4_25·b_1_24 + c_2_7·c_4_25·b_1_14 + c_2_72·b_1_33·b_3_13 + c_2_72·b_1_36 + c_2_72·b_1_23·b_3_11 + c_2_72·b_1_13·b_3_13 + c_2_72·b_1_33·a_3_6 + c_4_252·a_1_02 + c_4_24·c_4_25·a_1_02 + c_4_242·a_1_02 + c_2_72·c_4_25·b_1_32 + c_2_72·c_4_25·b_1_22 + c_2_73·b_1_1·b_1_33 + c_2_73·a_1_0·b_1_33 + c_2_74·b_1_12 + c_2_74·a_1_02
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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_2_7, a Duflot regular element of degree 2
- c_4_24, a Duflot regular element of degree 4
- c_4_25, a Duflot regular element of degree 4
- b_1_32 + b_1_22 + b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 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 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_1_3 → 0, an element of degree 1
- c_2_7 → c_1_22 + c_1_12, an element of degree 2
- a_3_6 → 0, an element of degree 3
- b_3_10 → 0, an element of degree 3
- b_3_11 → 0, an element of degree 3
- b_3_13 → 0, an element of degree 3
- b_3_14 → 0, an element of degree 3
- c_4_24 → c_1_24 + c_1_04, an element of degree 4
- c_4_25 → c_1_24, an element of degree 4
- b_5_40 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_7 → c_1_2·c_1_3 + c_1_22 + c_1_1·c_1_3 + c_1_12, an element of degree 2
- a_3_6 → 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_32 + c_1_12·c_1_3, an element of degree 3
- b_3_11 → c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_3_13 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_3_14 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_24 → c_1_22·c_1_32 + c_1_24 + c_1_0·c_1_33 + c_1_04, an element of degree 4
- c_4_25 → c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- b_5_40 → c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_1·c_1_2·c_1_33
+ c_1_1·c_1_22·c_1_32 + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_34 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- 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_1_3 → c_1_3, an element of degree 1
- c_2_7 → c_1_22 + c_1_12, an element of degree 2
- a_3_6 → 0, an element of degree 3
- b_3_10 → c_1_22·c_1_3, an element of degree 3
- b_3_11 → c_1_22·c_1_3, an element of degree 3
- b_3_13 → 0, an element of degree 3
- b_3_14 → c_1_22·c_1_3 + c_1_12·c_1_3, an element of degree 3
- c_4_24 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_02·c_1_32 + c_1_04, an element of degree 4
- c_4_25 → c_1_24, an element of degree 4
- b_5_40 → c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_12·c_1_33 + c_1_12·c_1_22·c_1_3, an element of degree 5
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_3, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- c_2_7 → c_1_2·c_1_3 + c_1_22 + c_1_1·c_1_3 + c_1_12, an element of degree 2
- a_3_6 → 0, an element of degree 3
- b_3_10 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_11 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_3_13 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- b_3_14 → c_1_33 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_24 → c_1_22·c_1_32 + c_1_24 + c_1_02·c_1_32 + c_1_04, an element of degree 4
- c_4_25 → c_1_34 + c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- b_5_40 → c_1_35 + c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_1·c_1_34 + c_1_14·c_1_3
+ c_1_0·c_1_34 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3, an element of degree 5
|