Simon King
David J. Green
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Cohomology of group number 936 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 1.
- It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2, 3 and 3, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 2.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
( − 2) · (t2 − t + 1) · (t2 + 1/2·t + 1/2) |
| (t − 1)3 · (t2 + 1)2 · (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:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- a_2_4, a nilpotent element of degree 2
- a_2_5, a nilpotent element of degree 2
- b_3_8, an element of degree 3
- b_4_11, an element of degree 4
- b_5_14, an element of degree 5
- b_5_15, an element of degree 5
- c_8_28, a Duflot regular element of degree 8
Ring relations
There are 21 minimal relations of maximal degree 10:
- b_1_0·b_1_1
- b_1_0·b_1_2
- b_1_12·b_1_2 + a_2_4·b_1_1
- a_2_5·b_1_2
- b_1_12·b_1_2 + a_2_5·b_1_1 + a_2_4·b_1_2
- a_2_4·b_1_1·b_1_2
- a_2_52 + a_2_4·a_2_5 + a_2_42
- b_1_0·b_3_8 + a_2_52 + a_2_42
- a_2_5·b_3_8 + a_2_4·b_3_8
- b_4_11·b_1_0
- b_1_0·b_5_14 + a_2_4·b_1_04 + a_2_42·b_1_02
- b_3_82 + b_1_2·b_5_15 + b_1_1·b_5_14 + b_1_13·b_3_8 + b_4_11·b_1_22
+ b_4_11·b_1_1·b_1_2 + b_4_11·b_1_12 + a_2_5·b_4_11 + a_2_4·b_4_11
- b_1_0·b_5_15 + a_2_5·b_1_04 + a_2_4·b_1_04 + a_2_42·b_1_02
- b_1_2·b_5_14 + b_1_1·b_5_15 + b_1_1·b_1_22·b_3_8 + b_1_13·b_3_8 + b_4_11·b_1_1·b_1_2
+ a_2_4·b_1_1·b_3_8
- a_2_5·b_5_15 + a_2_4·b_1_12·b_3_8 + a_2_4·b_4_11·b_1_2 + a_2_42·b_1_03
+ a_2_42·a_2_5·b_1_0
- a_2_5·b_5_14 + a_2_4·b_5_15 + a_2_4·b_5_14 + a_2_4·b_1_12·b_3_8 + a_2_4·b_4_11·b_1_2
+ a_2_42·a_2_5·b_1_0
- b_1_12·b_5_15 + b_1_14·b_3_8 + a_2_4·b_5_14 + a_2_4·b_1_12·b_3_8 + a_2_4·b_4_11·b_1_2
+ a_2_4·b_4_11·b_1_1 + a_2_42·b_1_03
- a_2_4·b_1_1·b_5_15 + a_2_4·b_1_13·b_3_8 + a_2_42·b_4_11
- b_5_152 + b_1_27·b_3_8 + b_1_1·b_1_26·b_3_8 + b_1_15·b_5_14 + b_1_17·b_3_8
+ b_4_11·b_1_2·b_5_15 + b_4_11·b_1_1·b_1_22·b_3_8 + b_4_11·b_1_1·b_1_25 + b_4_11·b_1_16 + b_4_112·b_1_22 + b_4_112·b_1_1·b_1_2 + a_2_5·b_4_112 + a_2_4·b_1_15·b_3_8 + a_2_4·b_4_11·b_1_1·b_3_8 + a_2_4·b_4_11·b_1_14 + a_2_4·a_2_5·b_1_06 + c_8_28·b_1_22
- b_5_14·b_5_15 + b_1_1·b_1_2·b_3_8·b_5_15 + b_1_1·b_1_26·b_3_8 + b_1_12·b_3_8·b_5_14
+ b_4_112·b_1_1·b_1_2 + a_2_5·b_4_112 + a_2_4·b_3_8·b_5_14 + a_2_4·b_1_13·b_5_14 + a_2_4·b_4_11·b_1_14 + a_2_4·b_4_112 + a_2_4·a_2_5·b_1_06 + a_2_42·b_1_06 + a_2_42·a_2_5·b_1_04 + c_8_28·b_1_1·b_1_2
- b_5_142 + b_1_17·b_3_8 + b_4_11·b_1_1·b_5_14 + b_4_112·b_1_12
+ a_2_4·b_1_13·b_5_14 + a_2_4·b_1_15·b_3_8 + a_2_4·b_4_11·b_1_1·b_3_8 + a_2_4·b_4_112 + a_2_42·b_1_06 + c_8_28·b_1_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 12.
- 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_8_28, a Duflot regular element of degree 8
- b_1_24 + b_1_14 + b_1_04 + b_4_11, an element of degree 4
- b_1_1·b_5_15 + b_1_13·b_3_8 + b_4_11·b_1_22 + b_4_11·b_1_1·b_1_2 + b_4_11·b_1_12, an element of degree 6
- The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 15].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
- We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 1 elements of degree 2.
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- b_1_0 → 0, an element of degree 1
- b_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
- a_2_5 → 0, an element of degree 2
- b_3_8 → 0, an element of degree 3
- b_4_11 → 0, an element of degree 4
- b_5_14 → 0, an element of degree 5
- b_5_15 → 0, an element of degree 5
- c_8_28 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- b_1_0 → c_1_1, an element of degree 1
- b_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
- a_2_5 → 0, an element of degree 2
- b_3_8 → 0, an element of degree 3
- b_4_11 → 0, an element of degree 4
- b_5_14 → 0, an element of degree 5
- b_5_15 → 0, an element of degree 5
- c_8_28 → c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_4 → 0, an element of degree 2
- a_2_5 → 0, an element of degree 2
- b_3_8 → c_1_1·c_1_22 + c_1_12·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
- b_4_11 → c_1_24 + c_1_12·c_1_22, an element of degree 4
- b_5_14 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_14 + c_1_04·c_1_1, an element of degree 5
- b_5_15 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_14 + c_1_02·c_1_13, an element of degree 5
- c_8_28 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24 + c_1_15·c_1_23
+ c_1_17·c_1_2 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_17 + c_1_04·c_1_24 + 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
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- a_2_4 → 0, an element of degree 2
- a_2_5 → 0, an element of degree 2
- b_3_8 → c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
- b_4_11 → c_1_24 + c_1_12·c_1_22, an element of degree 4
- b_5_14 → 0, an element of degree 5
- b_5_15 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- c_8_28 → c_1_28 + c_1_14·c_1_24 + c_1_0·c_1_17 + c_1_02·c_1_12·c_1_24
+ c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
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