Simon King
David J. Green
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Cohomology of group number 2183 of order 128
General information on the group
- The group has 5 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 + 2·t4 + 2·t2 + 2·t + 1 |
| (t + 1) · (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-6,-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 4:
- a_1_1, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- c_1_4, a Duflot regular element of degree 1
- a_4_22, a nilpotent element of degree 4
- b_4_32, an element of degree 4
- b_4_33, an element of degree 4
- b_4_34, an element of degree 4
- c_4_35, a Duflot regular element of degree 4
- c_4_36, a Duflot regular element of degree 4
Ring relations
There are 23 minimal relations of maximal degree 8:
- b_1_22 + b_1_0·b_1_2 + a_1_12
- b_1_22 + b_1_0·b_1_3 + a_1_1·b_1_2 + a_1_1·b_1_0 + a_1_12
- a_1_12·b_1_0
- a_1_1·b_1_2·b_1_3 + a_1_1·b_1_0·b_1_2
- a_4_22·a_1_1
- b_1_04·b_1_2 + b_4_32·b_1_2 + a_1_1·b_1_03·b_1_2 + a_4_22·b_1_2 + a_1_12·b_1_33
- a_1_1·b_1_04 + b_4_32·a_1_1 + a_4_22·b_1_2 + a_4_22·b_1_0 + a_1_12·b_1_33
- a_1_1·b_1_03·b_1_2 + a_1_1·b_1_04 + b_4_33·a_1_1 + a_4_22·b_1_0
- b_1_2·b_1_34 + b_1_04·b_1_2 + b_4_33·b_1_3 + b_4_33·b_1_2 + a_1_1·b_1_03·b_1_2
+ a_1_1·b_1_04 + a_4_22·b_1_3 + a_4_22·b_1_0
- b_1_04·b_1_2 + b_4_33·b_1_2 + b_4_33·b_1_0 + b_4_32·b_1_0 + a_1_1·b_1_03·b_1_2
+ a_4_22·b_1_2 + a_1_12·b_1_33
- b_1_2·b_1_34 + b_4_34·b_1_2 + b_4_33·b_1_2 + a_1_1·b_1_03·b_1_2 + a_4_22·b_1_3
+ a_4_22·b_1_2 + a_1_12·b_1_33
- b_1_2·b_1_34 + b_4_32·b_1_3 + b_4_34·a_1_1 + a_4_22·b_1_3 + a_4_22·b_1_2
+ a_1_12·b_1_33
- b_4_34·b_1_0 + b_4_33·b_1_2 + b_4_32·b_1_0 + a_1_1·b_1_03·b_1_2 + a_1_1·b_1_04
+ a_4_22·b_1_2 + a_1_12·b_1_33
- b_4_32·b_1_03·b_1_2 + b_4_322 + b_4_32·a_1_1·b_1_02·b_1_2 + a_4_22·b_1_04
+ c_4_36·b_1_03·b_1_2 + c_4_36·b_1_04 + c_4_35·b_1_03·b_1_2 + c_4_35·b_1_04
- b_4_32·a_1_1·b_1_02·b_1_2 + b_4_32·a_1_1·b_1_03 + a_4_22·b_1_03·b_1_2
+ a_4_22·b_4_32 + c_4_36·a_1_1·b_1_02·b_1_2 + c_4_36·a_1_1·b_1_03 + c_4_35·a_1_1·b_1_02·b_1_2 + c_4_35·a_1_1·b_1_03 + c_4_35·a_1_12·b_1_32
- b_4_332 + a_4_22·b_1_04 + a_1_12·b_1_36 + a_4_222 + c_4_36·b_1_04
+ c_4_35·b_1_04
- b_4_33·b_1_04 + b_4_32·b_1_03·b_1_2 + b_4_32·b_1_04 + b_4_32·b_4_33
+ a_4_22·b_1_04 + a_1_12·b_1_36 + a_4_222 + c_4_36·b_1_03·b_1_2 + c_4_36·b_1_04 + c_4_35·b_1_03·b_1_2 + c_4_35·b_1_04 + c_4_36·a_1_1·b_1_02·b_1_2 + c_4_35·a_1_1·b_1_02·b_1_2
- b_4_32·a_1_1·b_1_02·b_1_2 + b_4_32·a_1_1·b_1_03 + a_4_22·b_4_33
+ c_4_36·a_1_1·b_1_03 + c_4_35·a_1_1·b_1_03 + c_4_35·a_1_12·b_1_32
- a_1_12·b_1_36 + b_4_34·a_1_12·b_1_32 + a_4_222 + c_4_35·a_1_12·b_1_32
- b_4_33·b_1_04 + b_4_33·b_4_34 + b_4_32·b_1_34 + b_4_32·b_1_04
+ b_4_34·a_1_1·b_1_33 + b_4_32·a_1_1·b_1_03 + a_4_22·b_1_04 + a_1_12·b_1_36 + a_4_222 + c_4_36·b_1_04 + c_4_35·b_1_2·b_1_33 + c_4_35·b_1_03·b_1_2 + c_4_35·b_1_04
- b_4_33·b_1_04 + b_4_32·b_1_34 + b_4_32·b_1_03·b_1_2 + b_4_32·b_1_04
+ b_4_32·b_4_34 + b_4_32·a_1_1·b_1_03 + a_4_22·b_1_04 + a_1_12·b_1_36 + a_4_222 + c_4_36·b_1_03·b_1_2 + c_4_36·b_1_04 + c_4_35·b_1_2·b_1_33 + c_4_35·b_1_04 + c_4_36·a_1_1·b_1_02·b_1_2 + c_4_35·a_1_1·b_1_33
- b_4_34·b_1_34 + b_4_342 + b_4_33·b_1_04 + b_4_32·b_1_34 + b_4_32·b_1_04
+ b_4_34·a_1_1·b_1_33 + a_4_22·b_1_03·b_1_2 + a_4_22·b_1_04 + c_4_36·b_1_04 + c_4_35·b_1_34 + c_4_35·b_1_03·b_1_2 + c_4_35·b_1_04 + c_4_35·a_1_12·b_1_32
- b_4_32·a_1_1·b_1_02·b_1_2 + b_4_32·a_1_1·b_1_03 + a_4_22·b_1_03·b_1_2
+ a_4_22·b_4_34 + a_1_12·b_1_36 + a_4_222 + c_4_35·b_1_2·b_1_33 + c_4_35·b_1_03·b_1_2 + c_4_36·a_1_1·b_1_03 + c_4_35·a_1_1·b_1_03 + c_4_35·a_1_12·b_1_32
Data used for Benson′s test
- Benson′s completion test succeeded in degree 8.
- 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_4, a Duflot regular element of degree 1
- c_4_35, a Duflot regular element of degree 4
- c_4_36, a Duflot regular element of degree 4
- b_1_32 + b_1_22 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 3, 7].
- 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_1 → 0, an element of degree 1
- b_1_0 → 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_1_4 → c_1_0, an element of degree 1
- a_4_22 → 0, an element of degree 4
- b_4_32 → 0, an element of degree 4
- b_4_33 → 0, an element of degree 4
- b_4_34 → 0, an element of degree 4
- c_4_35 → c_1_24, an element of degree 4
- c_4_36 → c_1_24 + c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- b_1_0 → c_1_3, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_1_4 → c_1_0, an element of degree 1
- a_4_22 → 0, an element of degree 4
- b_4_32 → c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- b_4_33 → c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- b_4_34 → c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- c_4_35 → c_1_22·c_1_32 + c_1_24, an element of degree 4
- c_4_36 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- b_1_0 → 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_1_4 → c_1_0, an element of degree 1
- a_4_22 → 0, an element of degree 4
- b_4_32 → 0, an element of degree 4
- b_4_33 → 0, an element of degree 4
- b_4_34 → c_1_22·c_1_32, an element of degree 4
- c_4_35 → c_1_22·c_1_32 + c_1_24, an element of degree 4
- c_4_36 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- b_1_0 → 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_1_4 → c_1_0, an element of degree 1
- a_4_22 → 0, an element of degree 4
- b_4_32 → c_1_34, an element of degree 4
- b_4_33 → c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- b_4_34 → c_1_34 + c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- c_4_35 → c_1_22·c_1_32 + c_1_24, an element of degree 4
- c_4_36 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14, an element of degree 4
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