Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 1855 of order 128
General information on the group
- The group has 4 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
t4 − t3 + 1 |
| (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-4,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 6:
- a_1_0, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_3_10, an element of degree 3
- b_4_12, an element of degree 4
- c_4_13, a Duflot regular element of degree 4
- c_4_14, a Duflot regular element of degree 4
- b_5_23, an element of degree 5
- b_6_35, an element of degree 6
Ring relations
There are 22 minimal relations of maximal degree 12:
- a_1_0·b_1_1 + a_1_32
- a_1_0·b_1_2
- a_1_32·b_1_1 + a_1_0·a_1_32
- a_1_3·b_1_1·b_1_2 + a_1_3·b_1_12 + a_1_33 + a_1_03
- a_1_0·b_3_10
- b_4_12·a_1_0
- b_4_12·a_1_3
- b_3_102 + b_1_12·b_1_2·b_3_10 + b_4_12·b_1_22 + a_1_3·b_1_15 + c_4_13·b_1_22
- b_1_2·b_5_23 + b_1_1·b_1_22·b_3_10 + b_1_13·b_3_10 + b_4_12·b_1_12 + a_1_3·b_1_15
+ c_4_13·a_1_3·b_1_2
- a_1_0·b_5_23 + c_4_13·a_1_0·a_1_3 + c_4_13·a_1_02
- a_1_3·b_5_23 + c_4_13·a_1_32 + c_4_13·a_1_0·a_1_3
- b_1_14·b_3_10 + b_6_35·b_1_2 + b_4_12·b_3_10 + b_4_12·b_1_23 + b_4_12·b_1_1·b_1_22
+ b_4_12·b_1_13 + a_1_3·b_1_23·b_3_10 + a_1_3·b_1_13·b_3_10 + c_4_13·b_1_1·b_1_22 + c_4_14·a_1_3·b_1_12 + c_4_13·a_1_3·b_1_22 + c_4_14·a_1_33 + c_4_14·a_1_03
- b_6_35·a_1_0 + c_4_14·a_1_33 + c_4_14·a_1_02·a_1_3 + c_4_13·a_1_03
- a_1_3·b_1_13·b_3_10 + b_6_35·a_1_3 + c_4_13·a_1_3·b_1_12 + c_4_13·a_1_33
+ c_4_13·a_1_02·a_1_3 + c_4_13·a_1_03
- b_1_12·b_1_23·b_3_10 + b_6_35·b_1_22 + b_4_12·b_1_2·b_3_10 + b_4_12·b_1_24
+ b_4_12·b_1_1·b_1_23 + b_4_12·b_1_13·b_1_2 + b_4_122 + a_1_3·b_1_24·b_3_10 + b_6_35·a_1_3·b_1_1 + c_4_13·b_1_1·b_1_23 + c_4_14·a_1_3·b_1_13 + c_4_13·a_1_3·b_1_23 + c_4_13·a_1_3·b_1_13 + c_4_14·a_1_0·a_1_33 + c_4_14·a_1_02·a_1_32 + c_4_13·a_1_02·a_1_32
- b_3_10·b_5_23 + b_1_13·b_5_23 + b_1_13·b_1_22·b_3_10 + b_6_35·b_1_22
+ b_6_35·b_1_1·b_1_2 + b_6_35·b_1_12 + b_4_12·b_1_2·b_3_10 + b_4_12·b_1_24 + b_4_12·b_1_1·b_3_10 + b_4_12·b_1_1·b_1_23 + b_4_12·b_1_13·b_1_2 + b_4_12·b_1_14 + a_1_3·b_1_24·b_3_10 + a_1_3·b_1_17 + c_4_13·b_1_12·b_1_22 + c_4_14·a_1_3·b_1_13 + c_4_13·a_1_3·b_3_10 + c_4_13·a_1_3·b_1_23 + c_4_13·a_1_3·b_1_13 + c_4_14·a_1_0·a_1_33
- b_6_35·b_3_10 + b_6_35·b_1_23 + b_4_12·b_5_23 + b_4_12·b_1_25
+ b_4_12·b_1_1·b_1_24 + b_4_12·b_1_12·b_3_10 + b_4_12·b_1_13·b_1_22 + b_4_12·b_1_14·b_1_2 + b_4_122·b_1_2 + a_1_3·b_1_25·b_3_10 + a_1_3·b_1_18 + b_6_35·a_1_3·b_1_12 + c_4_13·b_1_1·b_1_2·b_3_10 + c_4_13·b_1_1·b_1_24 + c_4_13·b_1_14·b_1_2 + b_4_12·c_4_13·b_1_2 + c_4_14·a_1_3·b_1_1·b_3_10 + c_4_14·a_1_3·b_1_14 + c_4_13·a_1_3·b_1_2·b_3_10
- b_1_14·b_5_23 + b_6_35·b_3_10 + b_6_35·b_1_1·b_1_22 + b_6_35·b_1_12·b_1_2
+ b_6_35·b_1_13 + b_4_12·b_1_22·b_3_10 + b_4_12·b_1_1·b_1_24 + b_4_12·b_1_14·b_1_2 + b_4_12·b_1_15 + b_4_122·b_1_2 + c_4_13·b_1_1·b_1_2·b_3_10 + c_4_13·b_1_12·b_1_23 + c_4_13·b_1_13·b_1_22 + b_4_12·c_4_13·b_1_2 + c_4_14·a_1_3·b_1_1·b_3_10 + c_4_14·a_1_3·b_1_14 + c_4_13·a_1_3·b_1_2·b_3_10 + c_4_13·a_1_3·b_1_24 + c_4_13·a_1_3·b_1_14
- b_5_232 + b_4_12·b_1_12·b_1_24 + b_4_12·b_1_16 + b_4_122·b_1_12
+ c_4_13·b_1_12·b_1_24 + c_4_13·b_1_16 + c_4_132·a_1_32 + c_4_132·a_1_02
- b_4_12·b_1_1·b_5_23 + b_4_12·b_1_12·b_1_2·b_3_10 + b_4_12·b_1_12·b_1_24
+ b_4_12·b_1_14·b_1_22 + b_4_12·b_6_35 + b_4_122·b_1_22 + b_4_122·b_1_1·b_1_2 + b_4_122·b_1_12 + c_4_13·b_1_12·b_1_24 + c_4_13·b_1_14·b_1_22 + b_4_12·c_4_13·b_1_1·b_1_2
- b_6_35·b_5_23 + b_4_12·b_1_1·b_1_23·b_3_10 + b_4_12·b_1_12·b_1_22·b_3_10
+ b_4_12·b_1_13·b_1_24 + b_4_12·b_1_14·b_1_23 + b_4_12·b_1_16·b_1_2 + b_4_12·b_1_17 + b_4_12·b_6_35·b_1_1 + b_4_122·b_1_13 + b_6_35·a_1_3·b_1_14 + c_4_13·b_1_12·b_1_22·b_3_10 + c_4_13·b_1_13·b_1_24 + c_4_13·b_1_14·b_1_23 + c_4_13·b_1_16·b_1_2 + c_4_13·b_1_17 + c_4_13·b_6_35·b_1_2 + b_4_12·c_4_13·b_3_10 + b_4_12·c_4_13·b_1_23 + b_4_12·c_4_13·b_1_12·b_1_2 + b_4_12·c_4_13·b_1_13 + c_4_13·a_1_3·b_1_23·b_3_10 + c_4_132·b_1_1·b_1_22 + c_4_13·c_4_14·a_1_3·b_1_12 + c_4_132·a_1_3·b_1_22 + c_4_132·a_1_3·b_1_12 + c_4_13·c_4_14·a_1_02·a_1_3 + c_4_13·c_4_14·a_1_03 + c_4_132·a_1_33 + c_4_132·a_1_02·a_1_3
- b_6_35·b_1_14·b_1_22 + b_6_352 + b_4_12·b_1_15·b_1_23
+ b_4_12·b_1_16·b_1_22 + b_4_12·b_1_17·b_1_2 + b_4_12·b_1_18 + b_4_12·b_6_35·b_1_22 + b_4_122·b_1_2·b_3_10 + b_4_122·b_1_1·b_1_23 + b_4_122·b_1_12·b_1_22 + b_4_122·b_1_13·b_1_2 + b_4_122·b_1_14 + b_4_123 + a_1_3·b_1_111 + c_4_13·b_1_14·b_1_24 + c_4_13·b_1_15·b_1_23 + c_4_13·b_1_16·b_1_22 + c_4_13·b_1_18 + b_4_12·c_4_13·b_1_1·b_1_23 + b_4_122·c_4_13 + c_4_14·a_1_3·b_1_17 + c_4_13·a_1_3·b_1_17 + c_4_132·b_1_12·b_1_22
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_4_13, a Duflot regular element of degree 4
- c_4_14, a Duflot regular element of degree 4
- b_1_22 + b_1_1·b_1_2 + b_1_12, an element of degree 2
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 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_3 → 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_3_10 → 0, an element of degree 3
- b_4_12 → 0, an element of degree 4
- c_4_13 → c_1_04, an element of degree 4
- c_4_14 → c_1_14 + c_1_04, an element of degree 4
- b_5_23 → 0, an element of degree 5
- b_6_35 → 0, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_3 → 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
- b_3_10 → c_1_02·c_1_3, an element of degree 3
- b_4_12 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3, an element of degree 4
- c_4_13 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_14 → 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 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_04, an element of degree 4
- b_5_23 → c_1_0·c_1_23·c_1_3 + c_1_0·c_1_24 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_23, an element of degree 5
- b_6_35 → c_1_0·c_1_2·c_1_34 + c_1_0·c_1_22·c_1_33 + c_1_0·c_1_24·c_1_3 + c_1_0·c_1_25
+ c_1_02·c_1_23·c_1_3 + c_1_02·c_1_24 + c_1_03·c_1_2·c_1_32 + c_1_03·c_1_22·c_1_3 + c_1_04·c_1_2·c_1_3, an element of degree 6
|