Simon King
David J. Green
Cohomology
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Cohomology of group number 1853 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 3.
- Its center has rank 2.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all 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) · (t9 − t8 + t6 + 2·t5 + t4 + t3 + 2·t2 + 2·t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1)2 · (t4 + 1) |
- The a-invariants are -∞,-∞,-4,-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_2, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- a_3_10, a nilpotent element of degree 3
- b_4_12, an element of degree 4
- c_4_13, a Duflot regular element of degree 4
- b_5_18, an element of degree 5
- b_5_19, an element of degree 5
- c_8_33, a Duflot regular element of degree 8
Ring relations
There are 19 minimal relations of maximal degree 10:
- a_1_2·b_1_0
- b_1_0·b_1_1 + a_1_32
- a_1_32·b_1_1 + a_1_32·b_1_0
- a_1_3·b_1_12 + a_1_32·b_1_0 + a_1_2·a_1_3·b_1_1 + a_1_33 + a_1_23
- b_1_0·a_3_10
- a_1_23·b_1_12 + a_1_24·b_1_1 + a_1_25
- b_4_12·b_1_1 + a_1_3·b_1_1·a_3_10 + a_1_2·b_1_1·a_3_10 + a_1_22·b_1_13
+ a_1_22·a_3_10 + a_1_23·b_1_12 + a_1_25
- b_4_12·a_1_2 + a_1_3·b_1_1·a_3_10 + a_1_22·a_3_10 + a_1_23·b_1_12 + a_1_25
- a_3_102 + a_1_2·b_1_12·a_3_10 + a_1_22·b_1_14 + a_1_2·a_1_3·b_1_1·a_3_10
+ a_1_22·b_1_1·a_3_10 + a_1_25·b_1_1 + c_4_13·a_1_22
- a_1_2·b_5_18 + a_3_102 + a_1_2·b_1_12·a_3_10 + a_1_22·b_1_14
+ a_1_22·b_1_1·a_3_10 + a_1_22·a_1_3·a_3_10 + a_1_23·a_3_10 + c_4_13·a_1_2·b_1_1
- b_1_0·b_5_19 + b_1_0·b_5_18 + a_1_3·b_5_18 + c_4_13·a_1_3·b_1_1 + c_4_13·a_1_32
+ c_4_13·a_1_2·a_1_3
- b_1_1·b_5_18 + a_1_3·b_5_19 + a_1_3·b_5_18 + a_1_2·a_1_3·b_1_1·a_3_10
+ a_1_22·a_1_3·a_3_10 + a_1_23·a_3_10 + c_4_13·b_1_12 + c_4_13·a_1_3·b_1_1 + c_4_13·a_1_2·b_1_1 + c_4_13·a_1_2·a_1_3
- b_4_12·a_3_10 + a_1_22·a_1_3·b_1_1·a_3_10 + a_1_24·a_3_10 + c_4_13·a_1_2·a_1_3·b_1_1
+ c_4_13·a_1_23
- a_1_3·b_1_1·b_5_19 + a_1_32·b_5_19 + a_1_22·a_1_3·b_1_1·a_3_10
+ a_1_23·b_1_1·a_3_10
- a_3_10·b_5_18 + c_4_13·b_1_1·a_3_10 + c_4_13·a_1_2·a_3_10 + c_4_13·a_1_22·a_1_3·b_1_1
+ c_4_13·a_1_23·a_1_3 + c_4_13·a_1_24
- b_1_03·b_5_18 + b_4_122 + b_4_12·a_1_3·b_1_03 + a_1_25·a_3_10 + c_4_13·b_1_04
+ c_4_13·a_1_24
- b_5_182 + b_4_122·b_1_02 + b_4_12·a_1_3·b_1_05 + b_4_122·a_1_3·b_1_0
+ c_8_33·b_1_02 + c_4_13·b_1_06 + b_4_12·c_4_13·b_1_02 + c_4_13·a_1_3·b_1_05 + c_4_132·b_1_12 + c_4_132·a_1_22
- b_5_18·b_5_19 + b_5_182 + b_4_122·a_1_3·b_1_0 + c_4_13·b_1_1·b_5_19
+ c_8_33·a_1_3·b_1_0 + c_4_13·a_1_3·b_1_05 + c_4_13·a_1_2·b_5_19 + b_4_12·c_4_13·a_1_3·b_1_0 + c_4_132·b_1_12 + c_4_132·a_1_22
- b_5_192 + b_5_182 + b_1_17·a_3_10 + a_1_2·b_1_16·a_3_10 + c_4_13·b_1_16
+ c_8_33·a_1_32 + c_4_13·a_1_22·b_1_14 + c_4_13·a_1_25·b_1_1 + c_4_132·b_1_12 + c_4_132·a_1_22
Data used for Benson′s test
- Benson′s completion test succeeded in degree 11.
- However, the last relation was already found in degree 10 and the last generator in degree 8.
- The following is a filter regular homogeneous system of parameters:
- c_4_13, a Duflot regular element of degree 4
- c_8_33, a Duflot regular element of degree 8
- b_1_12 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 11].
- 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_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- a_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
- b_5_18 → 0, an element of degree 5
- b_5_19 → 0, an element of degree 5
- c_8_33 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → 0, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_4_12 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- c_4_13 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_18 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- b_5_19 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- c_8_33 → c_1_1·c_1_27 + c_1_18 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25
+ c_1_02·c_1_12·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_4_12 → 0, an element of degree 4
- c_4_13 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_18 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- b_5_19 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- c_8_33 → c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_08, an element of degree 8
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