Simon King
David J. Green
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Cohomology of group number 2170 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 5.
- Its center has rank 4.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 5.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 4.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t4 − t3 − 1 |
| (t − 1)5 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-∞,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 4:
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- c_1_3, a Duflot regular element of degree 1
- c_1_4, a Duflot regular element of degree 1
- a_3_24, a nilpotent element of degree 3
- b_3_23, an element of degree 3
- b_3_25, an element of degree 3
- c_4_45, a Duflot regular element of degree 4
- c_4_46, a Duflot regular element of degree 4
Ring relations
There are 14 minimal relations of maximal degree 6:
- a_1_1·b_1_0 + a_1_22 + a_1_12
- a_1_2·b_1_0 + a_1_12
- a_1_12·a_1_2
- a_1_1·a_1_22 + a_1_13
- b_1_0·a_3_24 + a_1_1·b_3_23
- a_1_2·b_3_23 + a_1_1·a_3_24
- b_1_0·a_3_24 + a_1_1·b_3_25 + a_1_2·a_3_24
- b_1_0·a_3_24 + a_1_2·b_3_25
- a_1_22·a_3_24 + a_1_1·a_1_2·a_3_24 + a_1_12·a_3_24
- b_3_252 + b_3_232 + a_3_24·b_3_23 + a_3_242 + c_4_45·b_1_02 + c_4_45·a_1_22
+ c_4_45·a_1_1·a_1_2 + c_4_45·a_1_12
- b_3_252 + b_3_232 + a_3_24·b_3_23 + c_4_45·b_1_02 + c_4_46·a_1_12
+ c_4_45·a_1_1·a_1_2 + c_4_45·a_1_12
- b_3_232 + c_4_46·b_1_02 + c_4_45·a_1_12
- a_3_24·b_3_25 + a_3_24·b_3_23 + c_4_46·a_1_1·a_1_2 + c_4_45·a_1_1·a_1_2
+ c_4_45·a_1_12
- b_3_252 + b_3_232 + c_4_45·b_1_02 + c_4_46·a_1_22 + c_4_45·a_1_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 9.
- However, the last relation was already found in degree 6 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_1_3, a Duflot regular element of degree 1
- c_1_4, a Duflot regular element of degree 1
- c_4_45, a Duflot regular element of degree 4
- c_4_46, a Duflot regular element of degree 4
- b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 5, 7].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 4
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- c_1_4 → c_1_1, an element of degree 1
- a_3_24 → 0, an element of degree 3
- b_3_23 → 0, an element of degree 3
- b_3_25 → 0, an element of degree 3
- c_4_45 → c_1_34, an element of degree 4
- c_4_46 → c_1_34 + c_1_24, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 5
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → c_1_4, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- c_1_4 → c_1_1, an element of degree 1
- a_3_24 → 0, an element of degree 3
- b_3_23 → c_1_3·c_1_42 + c_1_32·c_1_4 + c_1_2·c_1_42 + c_1_22·c_1_4, an element of degree 3
- b_3_25 → c_1_2·c_1_42 + c_1_22·c_1_4, an element of degree 3
- c_4_45 → c_1_32·c_1_42 + c_1_34, an element of degree 4
- c_4_46 → c_1_32·c_1_42 + c_1_34 + c_1_22·c_1_42 + c_1_24, an element of degree 4
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