Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 278 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 3.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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) · (t5 + t2 + 1) |
| (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 8:
- 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
- c_2_5, a Duflot regular element of degree 2
- a_3_8, a nilpotent element of degree 3
- a_5_15, a nilpotent element of degree 5
- a_5_16, a nilpotent element of degree 5
- a_6_17, a nilpotent element of degree 6
- c_8_37, a Duflot regular element of degree 8
Ring relations
There are 27 minimal relations of maximal degree 12:
- a_1_02
- a_1_0·a_1_1
- a_1_0·b_1_22 + a_1_13
- a_2_4·a_1_0
- a_2_42 + a_2_4·a_1_12
- a_2_4·b_1_22 + a_1_1·a_3_8 + a_1_13·b_1_2 + a_2_4·a_1_12
- a_1_0·a_3_8 + a_2_4·a_1_12
- a_1_13·b_1_22
- a_2_4·a_3_8 + a_1_12·a_3_8 + a_2_4·a_1_12·b_1_2
- a_3_82 + a_1_1·b_1_22·a_3_8 + a_1_12·b_1_24
- a_1_0·a_5_15
- a_1_1·a_5_16 + a_1_1·a_5_15 + a_1_13·a_3_8 + c_2_5·a_1_1·a_3_8 + c_2_5·a_1_13·b_1_2
+ a_2_4·c_2_5·a_1_12
- a_1_0·a_5_16 + a_1_13·a_3_8 + a_2_4·c_2_5·a_1_12
- b_1_22·a_5_16 + b_1_22·a_5_15 + a_1_12·a_5_15 + a_1_12·b_1_22·a_3_8
+ a_1_13·b_1_2·a_3_8 + c_2_5·b_1_22·a_3_8 + c_2_5·a_1_12·b_1_23 + c_2_5·a_1_12·a_3_8
- a_2_4·a_5_16 + a_2_4·a_5_15 + c_2_5·a_1_12·a_3_8
- a_1_1·b_1_2·a_5_15 + a_1_1·b_1_23·a_3_8 + a_6_17·a_1_1 + a_2_4·a_5_15
+ a_1_12·b_1_22·a_3_8 + c_2_5·a_1_1·b_1_2·a_3_8 + c_2_5·a_1_12·b_1_23 + c_2_5·a_1_12·a_3_8 + a_2_4·c_2_5·a_1_12·b_1_2 + c_2_52·a_1_13
- a_6_17·a_1_0 + a_1_13·b_1_2·a_3_8 + a_2_4·c_2_5·a_1_12·b_1_2
- a_3_8·a_5_16 + a_3_8·a_5_15 + a_2_4·a_1_1·a_5_15 + c_2_5·a_1_1·b_1_22·a_3_8
+ c_2_5·a_1_12·b_1_24 + c_2_5·a_1_12·b_1_2·a_3_8
- b_1_23·a_5_15 + b_1_25·a_3_8 + a_6_17·b_1_22 + a_3_8·a_5_15 + a_1_1·b_1_24·a_3_8
+ a_1_12·b_1_26 + a_1_12·b_1_23·a_3_8 + a_6_17·a_1_12 + c_2_5·b_1_23·a_3_8 + c_2_5·a_1_1·b_1_25 + c_2_5·a_1_1·b_1_22·a_3_8 + c_2_5·a_1_12·b_1_24 + c_2_52·a_1_12·b_1_22 + c_2_52·a_1_13·b_1_2
- a_2_4·b_1_2·a_5_15 + a_2_4·a_6_17 + a_1_12·b_1_23·a_3_8 + a_2_4·a_1_1·a_5_15
+ c_2_5·a_1_13·a_3_8 + a_2_4·c_2_52·a_1_12
- b_1_2·a_3_8·a_5_15 + a_1_1·b_1_25·a_3_8 + a_1_12·b_1_27 + a_6_17·a_3_8
+ a_1_1·a_3_8·a_5_15 + a_1_12·b_1_24·a_3_8 + a_6_17·a_1_12·b_1_2 + c_2_5·a_1_12·b_1_25 + c_2_5·a_1_12·b_1_22·a_3_8 + c_2_5·a_1_13·b_1_2·a_3_8 + c_2_52·a_1_12·a_3_8 + a_2_4·c_2_52·a_1_12·b_1_2
- a_5_162 + a_5_152 + c_2_52·a_1_1·b_1_22·a_3_8 + c_2_52·a_1_12·b_1_24
- a_5_15·a_5_16 + a_5_152 + a_1_12·a_3_8·a_5_15 + c_2_5·a_3_8·a_5_15
+ c_2_5·a_1_12·b_1_23·a_3_8 + c_2_5·a_6_17·a_1_12 + c_2_52·a_1_12·b_1_2·a_3_8 + c_2_52·a_1_13·a_3_8
- a_5_152 + a_1_1·b_1_26·a_3_8 + a_1_12·b_1_28 + a_6_17·a_1_1·b_1_23
+ a_6_17·a_1_1·a_3_8 + a_1_12·a_3_8·a_5_15 + c_8_37·a_1_12 + c_2_5·a_1_1·b_1_24·a_3_8 + c_2_5·a_1_12·b_1_26 + c_2_5·a_1_12·b_1_23·a_3_8 + a_2_4·c_2_5·a_1_1·a_5_15 + c_2_52·a_1_12·b_1_24 + c_2_53·a_1_12·b_1_22 + c_2_53·a_1_13·b_1_2 + a_2_4·c_2_53·a_1_12
- a_6_17·a_5_16 + a_6_17·a_5_15 + c_2_5·a_6_17·a_3_8 + c_2_5·a_6_17·a_1_12·b_1_2
- a_6_17·a_5_15 + a_6_17·b_1_22·a_3_8 + a_6_17·a_1_1·b_1_24 + a_6_17·a_1_12·a_3_8
+ c_8_37·a_1_12·b_1_2 + c_2_5·a_1_1·b_1_25·a_3_8 + c_2_5·a_1_12·b_1_27 + c_2_5·a_6_17·a_3_8 + c_2_5·a_6_17·a_1_1·b_1_22 + a_2_4·c_8_37·a_1_1 + c_2_5·a_1_1·a_3_8·a_5_15 + a_2_4·c_2_5·a_6_17·a_1_1 + c_2_52·a_1_1·b_1_23·a_3_8 + c_2_52·a_1_12·b_1_25 + c_2_52·a_1_12·a_5_15 + c_2_52·a_1_12·b_1_22·a_3_8 + c_2_52·a_1_13·b_1_2·a_3_8 + c_2_53·a_1_12·b_1_23 + a_2_4·c_2_53·a_1_12·b_1_2
- a_6_17·a_1_1·b_1_25 + a_6_172 + a_6_17·a_1_1·b_1_22·a_3_8
+ c_8_37·a_1_12·b_1_22 + c_2_5·a_1_1·b_1_26·a_3_8 + c_2_5·a_1_12·b_1_28 + c_2_5·a_1_12·b_1_25·a_3_8 + a_2_4·c_8_37·a_1_12 + c_2_5·a_1_12·a_3_8·a_5_15 + c_2_52·a_1_1·b_1_24·a_3_8 + c_2_52·a_1_12·b_1_26 + c_2_53·a_1_12·b_1_24
Data used for Benson′s test
- Benson′s completion test succeeded in degree 12.
- 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_5, a Duflot regular element of degree 2
- c_8_37, a Duflot regular element of degree 8
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 9].
- 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
- 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
- c_2_5 → c_1_12, an element of degree 2
- a_3_8 → 0, an element of degree 3
- a_5_15 → 0, an element of degree 5
- a_5_16 → 0, an element of degree 5
- a_6_17 → 0, an element of degree 6
- c_8_37 → c_1_18 + 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
- 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
- c_2_5 → c_1_12, an element of degree 2
- a_3_8 → 0, an element of degree 3
- a_5_15 → 0, an element of degree 5
- a_5_16 → 0, an element of degree 5
- a_6_17 → 0, an element of degree 6
- c_8_37 → c_1_16·c_1_22 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
|