Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 1756 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 2.
- It has 5 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 exceeds the Duflot bound, which is 2.
- The Poincaré series is
t7 + 2·t5 + t3 + t2 + t + 1 |
| (t + 1) · (t − 1)4 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-∞,-5,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 11 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
- c_1_3, a Duflot regular element of degree 1
- b_2_8, an element of degree 2
- b_2_9, an element of degree 2
- b_3_19, an element of degree 3
- b_5_50, an element of degree 5
- b_5_52, an element of degree 5
- b_5_53, an element of degree 5
- c_8_153, a Duflot regular element of degree 8
Ring relations
There are 27 minimal relations of maximal degree 10:
- a_1_0·b_1_1
- a_1_0·b_1_2
- a_1_03
- b_2_9·b_1_1 + b_2_8·b_1_2
- b_2_8·b_1_22 + b_2_8·b_1_1·b_1_2
- b_1_2·b_3_19 + b_2_9·a_1_02
- b_1_1·b_3_19 + b_2_8·a_1_02
- b_2_8·b_2_9·b_1_2 + b_2_82·b_1_2
- a_1_02·b_3_19
- a_1_0·b_5_50
- b_1_2·b_5_52 + b_2_92·b_1_22 + b_2_8·b_1_13·b_1_2 + b_2_82·b_1_1·b_1_2
+ b_2_8·b_2_9·a_1_02
- b_3_192 + b_2_8·b_2_92 + b_2_82·b_2_9 + a_1_0·b_5_52 + b_2_9·a_1_0·b_3_19
+ b_2_8·a_1_0·b_3_19 + b_2_8·b_2_9·a_1_02
- b_1_1·b_5_52 + b_2_8·b_1_13·b_1_2 + b_2_82·b_1_1·b_1_2 + b_2_82·b_1_12
+ b_2_8·b_2_9·a_1_02
- b_3_192 + b_2_8·b_2_92 + b_2_82·b_2_9 + a_1_0·b_5_53
- b_1_2·b_5_53 + b_1_2·b_5_50 + b_1_1·b_5_53 + b_2_92·b_1_22 + b_2_8·b_1_13·b_1_2
+ b_2_82·b_1_1·b_1_2 + b_2_82·b_1_12 + b_2_92·a_1_02 + b_2_82·a_1_02
- a_1_02·b_5_52
- b_2_9·b_5_53 + b_2_9·b_5_52 + b_2_9·b_5_50 + b_2_92·b_3_19 + b_2_8·b_5_53 + b_2_8·b_5_52
+ b_2_82·b_3_19 + b_2_82·b_1_12·b_1_2 + b_2_83·b_1_2
- b_2_8·b_1_2·b_5_50 + b_2_82·b_1_13·b_1_2 + b_2_83·b_1_1·b_1_2
- b_3_19·b_5_50 + b_2_9·a_1_0·b_5_52 + b_2_8·a_1_0·b_5_52 + b_2_82·b_2_9·a_1_02
- b_3_19·b_5_53 + b_3_19·b_5_52 + b_3_19·b_5_50 + b_2_8·b_2_93 + b_2_83·b_2_9
+ b_2_9·a_1_0·b_5_52 + b_2_92·a_1_0·b_3_19 + b_2_82·a_1_0·b_3_19
- b_2_8·b_2_9·b_5_50 + b_2_83·b_1_12·b_1_2 + b_2_84·b_1_2 + a_1_0·b_3_19·b_5_52
- b_5_50·b_5_52 + b_2_92·b_1_2·b_5_50 + b_2_82·b_1_1·b_5_50 + b_2_82·b_1_15·b_1_2
+ b_2_83·b_1_13·b_1_2 + b_2_8·b_2_92·a_1_0·b_3_19 + b_2_82·b_2_9·a_1_0·b_3_19 + b_2_83·b_2_9·a_1_02
- b_5_52·b_5_53 + b_5_522 + b_2_9·b_3_19·b_5_52 + b_2_92·b_1_2·b_5_50
+ b_2_8·b_3_19·b_5_52 + b_2_8·b_1_13·b_5_53 + b_2_82·b_1_15·b_1_2 + b_2_83·b_1_14 + b_2_84·b_1_1·b_1_2 + b_2_8·b_2_92·a_1_0·b_3_19 + b_2_83·b_2_9·a_1_02
- b_5_532 + b_5_522 + b_1_25·b_5_50 + b_1_12·b_1_23·b_5_50 + b_1_14·b_1_2·b_5_50
+ b_1_15·b_5_53 + b_2_92·b_1_26 + b_2_8·b_2_94 + b_2_82·b_1_16 + b_2_82·b_2_93 + b_2_83·b_2_92 + b_2_84·b_2_9 + b_2_92·a_1_0·b_5_52 + b_2_93·a_1_0·b_3_19 + b_2_8·b_2_92·a_1_0·b_3_19 + b_2_82·a_1_0·b_5_52 + b_2_82·b_2_9·a_1_0·b_3_19 + b_2_83·a_1_0·b_3_19 + b_2_83·b_2_9·a_1_02 + c_8_153·b_1_22
- b_5_532 + b_5_522 + b_5_50·b_5_53 + b_1_1·b_1_24·b_5_50 + b_1_13·b_1_22·b_5_50
+ b_1_15·b_5_53 + b_2_92·b_1_2·b_5_50 + b_2_8·b_1_13·b_5_53 + b_2_8·b_1_17·b_1_2 + b_2_8·b_2_94 + b_2_82·b_1_1·b_5_53 + b_2_82·b_1_1·b_5_50 + b_2_82·b_1_16 + b_2_82·b_2_93 + b_2_83·b_1_13·b_1_2 + b_2_83·b_1_14 + b_2_83·b_2_92 + b_2_84·b_1_1·b_1_2 + b_2_84·b_1_12 + b_2_84·b_2_9 + b_2_93·a_1_0·b_3_19 + b_2_83·a_1_0·b_3_19 + b_2_84·a_1_02 + c_8_153·b_1_1·b_1_2
- b_5_522 + b_2_94·b_1_22 + b_2_82·b_1_15·b_1_2 + b_2_82·b_2_93
+ b_2_83·b_2_92 + b_2_84·b_1_12 + b_2_8·b_2_9·a_1_0·b_5_52 + b_2_83·b_2_9·a_1_02 + b_2_84·a_1_02 + c_8_153·a_1_02
- b_5_532 + b_5_522 + b_5_502 + b_1_12·b_1_23·b_5_50 + b_1_14·b_1_2·b_5_50
+ b_1_15·b_5_53 + b_1_15·b_5_50 + b_2_8·b_1_13·b_5_50 + b_2_8·b_2_94 + b_2_82·b_2_93 + b_2_83·b_1_13·b_1_2 + b_2_83·b_2_92 + b_2_84·b_1_12 + b_2_84·b_2_9 + b_2_92·a_1_0·b_5_52 + b_2_93·a_1_0·b_3_19 + b_2_8·b_2_92·a_1_0·b_3_19 + b_2_82·a_1_0·b_5_52 + b_2_82·b_2_9·a_1_0·b_3_19 + b_2_83·a_1_0·b_3_19 + b_2_83·b_2_9·a_1_02 + c_8_153·b_1_12
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_1_3, a Duflot regular element of degree 1
- c_8_153, a Duflot regular element of degree 8
- b_1_24 + b_1_12·b_1_22 + b_1_14 + b_2_92 + b_2_8·b_2_9 + b_2_82, an element of degree 4
- b_3_19 + b_1_1·b_1_22 + b_1_12·b_1_2 + b_2_9·b_1_2 + b_2_8·b_1_2 + b_2_8·b_1_1, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 12].
- 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 2 terms of the above HSOP, together with 2 elements of degree 2.
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
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- b_2_9 → 0, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_5_50 → 0, an element of degree 5
- b_5_52 → 0, an element of degree 5
- b_5_53 → 0, an element of degree 5
- c_8_153 → c_1_18, 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_2, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- b_2_9 → 0, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_5_50 → c_1_1·c_1_2·c_1_33 + c_1_1·c_1_23·c_1_3 + c_1_12·c_1_33 + c_1_12·c_1_23
+ c_1_14·c_1_3 + c_1_14·c_1_2, an element of degree 5
- b_5_52 → 0, an element of degree 5
- b_5_53 → 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_14·c_1_3, an element of degree 5
- c_8_153 → c_1_1·c_1_2·c_1_36 + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_36
+ c_1_12·c_1_23·c_1_33 + c_1_12·c_1_26 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_23·c_1_3 + c_1_18, 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_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- b_2_9 → 0, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_5_50 → 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, an element of degree 5
- b_5_52 → c_1_2·c_1_34 + c_1_23·c_1_32, an element of degree 5
- b_5_53 → c_1_2·c_1_34 + c_1_23·c_1_32, an element of degree 5
- c_8_153 → c_1_38 + c_1_26·c_1_32 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_26·c_1_3
+ c_1_12·c_1_26 + c_1_14·c_1_34 + c_1_14·c_1_23·c_1_3 + c_1_18, 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 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- b_2_9 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_5_50 → 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, an element of degree 5
- b_5_52 → c_1_2·c_1_34 + c_1_23·c_1_32, an element of degree 5
- b_5_53 → c_1_2·c_1_34 + c_1_23·c_1_32 + 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, an element of degree 5
- c_8_153 → c_1_24·c_1_34 + c_1_26·c_1_32 + c_1_1·c_1_25·c_1_32 + c_1_1·c_1_26·c_1_3
+ c_1_12·c_1_22·c_1_34 + c_1_12·c_1_25·c_1_3 + c_1_12·c_1_26 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32 + c_1_18, 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 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → c_1_22, an element of degree 2
- b_2_9 → c_1_32, an element of degree 2
- b_3_19 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_5_50 → 0, an element of degree 5
- b_5_52 → c_1_22·c_1_33 + c_1_23·c_1_32, an element of degree 5
- b_5_53 → c_1_2·c_1_34 + c_1_24·c_1_3, an element of degree 5
- c_8_153 → c_1_22·c_1_36 + c_1_24·c_1_34 + c_1_28 + c_1_12·c_1_22·c_1_34
+ c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32 + c_1_14·c_1_24 + c_1_18, 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_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- b_2_8 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- b_2_9 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- b_3_19 → 0, an element of degree 3
- b_5_50 → c_1_2·c_1_34 + c_1_24·c_1_3, an element of degree 5
- b_5_52 → c_1_23·c_1_32 + c_1_24·c_1_3, an element of degree 5
- b_5_53 → c_1_2·c_1_34 + c_1_23·c_1_32 + 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, an element of degree 5
- c_8_153 → c_1_38 + c_1_27·c_1_3 + c_1_1·c_1_25·c_1_32 + c_1_1·c_1_26·c_1_3
+ c_1_12·c_1_22·c_1_34 + c_1_12·c_1_25·c_1_3 + c_1_12·c_1_26 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32 + c_1_18, an element of degree 8
|