Simon King
David J. Green
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Cohomology of group number 2248 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 3.
- Its center has rank 2.
- It has 5 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) · (t2 + t + 1) · (t6 + t5 + t2 + t + 1) |
| (t − 1)3 · (t2 + 1)2 · (t4 + 1) |
- The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 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
- b_1_3, an element of degree 1
- b_1_4, an element of degree 1
- c_4_31, a Duflot regular element of degree 4
- b_5_39, an element of degree 5
- b_5_40, an element of degree 5
- c_8_87, a Duflot regular element of degree 8
Ring relations
There are 11 minimal relations of maximal degree 10:
- b_1_42 + b_1_0·b_1_2
- b_1_32 + b_1_1·b_1_3 + b_1_12 + b_1_0·b_1_4 + b_1_0·b_1_1
- b_1_0·b_1_22 + b_1_02·b_1_2
- b_1_13 + b_1_0·b_1_22 + b_1_0·b_1_1·b_1_4 + b_1_02·b_1_4 + b_1_02·b_1_1
- b_1_03·b_1_2·b_1_4 + b_1_04·b_1_2
- b_1_4·b_5_40 + b_1_4·b_5_39 + b_1_2·b_5_40 + c_4_31·b_1_2·b_1_4 + c_4_31·b_1_2·b_1_3
+ c_4_31·b_1_1·b_1_4 + c_4_31·b_1_1·b_1_2 + c_4_31·b_1_0·b_1_2
- b_1_4·b_5_40 + b_1_0·b_5_40 + b_1_0·b_5_39 + b_1_02·b_1_12·b_1_2·b_1_4
+ b_1_03·b_1_12·b_1_4 + b_1_04·b_1_3·b_1_4 + b_1_04·b_1_2·b_1_3 + c_4_31·b_1_3·b_1_4 + c_4_31·b_1_1·b_1_4 + c_4_31·b_1_0·b_1_4 + c_4_31·b_1_0·b_1_2 + c_4_31·b_1_0·b_1_1
- b_1_0·b_1_2·b_5_39 + c_4_31·b_1_0·b_1_2·b_1_4 + c_4_31·b_1_0·b_1_2·b_1_3
+ c_4_31·b_1_02·b_1_2
- b_5_402 + b_5_39·b_5_40 + b_1_24·b_1_3·b_5_40 + b_1_25·b_5_40
+ b_1_1·b_1_23·b_1_3·b_5_40 + b_1_1·b_1_24·b_5_40 + b_1_12·b_1_22·b_1_3·b_5_40 + b_1_0·b_1_12·b_1_2·b_1_3·b_5_40 + b_1_02·b_1_12·b_1_2·b_5_40 + b_1_03·b_1_2·b_1_3·b_5_40 + b_1_03·b_1_1·b_1_3·b_5_40 + b_1_03·b_1_1·b_1_3·b_5_39 + b_1_03·b_1_1·b_1_2·b_5_40 + b_1_03·b_1_12·b_5_40 + b_1_03·b_1_12·b_5_39 + b_1_04·b_1_2·b_5_40 + b_1_07·b_1_1·b_1_3·b_1_4 + b_1_08·b_1_1·b_1_4 + c_8_87·b_1_2·b_1_4 + c_8_87·b_1_0·b_1_4 + c_4_31·b_1_3·b_5_40 + c_4_31·b_1_3·b_5_39 + c_4_31·b_1_2·b_5_40 + c_4_31·b_1_24·b_1_3·b_1_4 + c_4_31·b_1_25·b_1_3 + c_4_31·b_1_1·b_5_39 + c_4_31·b_1_1·b_1_24·b_1_4 + c_4_31·b_1_1·b_1_24·b_1_3 + c_4_31·b_1_1·b_1_25 + c_4_31·b_1_12·b_1_23·b_1_4 + c_4_31·b_1_0·b_5_40 + c_4_31·b_1_0·b_5_39 + c_4_31·b_1_02·b_1_12·b_1_2·b_1_4 + c_4_31·b_1_02·b_1_12·b_1_2·b_1_3 + c_4_31·b_1_03·b_1_1·b_1_3·b_1_4 + c_4_31·b_1_03·b_1_12·b_1_4 + c_4_31·b_1_03·b_1_12·b_1_3 + c_4_31·b_1_04·b_1_3·b_1_4 + c_4_31·b_1_04·b_1_2·b_1_3 + c_4_31·b_1_04·b_1_1·b_1_4 + c_4_31·b_1_04·b_1_1·b_1_2 + c_4_31·b_1_05·b_1_2 + c_4_31·b_1_05·b_1_1 + c_4_312·b_1_2·b_1_3 + c_4_312·b_1_1·b_1_3 + c_4_312·b_1_1·b_1_2 + c_4_312·b_1_12 + c_4_312·b_1_0·b_1_4 + c_4_312·b_1_0·b_1_2 + c_4_312·b_1_0·b_1_1
- b_5_402 + b_5_392 + b_1_24·b_1_3·b_5_40 + b_1_24·b_1_3·b_5_39 + b_1_25·b_5_39
+ b_1_1·b_1_23·b_1_3·b_5_40 + b_1_1·b_1_23·b_1_3·b_5_39 + b_1_1·b_1_24·b_5_40 + b_1_1·b_1_24·b_5_39 + b_1_12·b_1_22·b_1_3·b_5_40 + b_1_12·b_1_22·b_1_3·b_5_39 + b_1_12·b_1_23·b_5_40 + b_1_0·b_1_12·b_1_2·b_1_3·b_5_40 + b_1_02·b_1_1·b_1_2·b_1_3·b_5_40 + b_1_02·b_1_12·b_1_2·b_5_40 + b_1_03·b_1_2·b_1_3·b_5_40 + b_1_03·b_1_1·b_1_2·b_5_40 + b_1_07·b_1_1·b_1_2·b_1_3 + b_1_08·b_1_1·b_1_2 + c_8_87·b_1_22 + c_8_87·b_1_0·b_1_2 + c_4_31·b_1_24·b_1_3·b_1_4 + c_4_31·b_1_25·b_1_3 + c_4_31·b_1_26 + c_4_31·b_1_1·b_1_23·b_1_3·b_1_4 + c_4_31·b_1_12·b_1_23·b_1_4 + c_4_31·b_1_12·b_1_23·b_1_3 + c_4_31·b_1_0·b_1_12·b_1_2·b_1_3·b_1_4 + c_4_31·b_1_02·b_1_1·b_1_2·b_1_3·b_1_4 + c_4_31·b_1_02·b_1_12·b_1_2·b_1_4 + c_4_31·b_1_02·b_1_12·b_1_2·b_1_3 + c_4_31·b_1_03·b_1_12·b_1_2 + c_4_31·b_1_04·b_1_2·b_1_3 + c_4_31·b_1_05·b_1_2 + c_4_312·b_1_22 + c_4_312·b_1_12 + c_4_312·b_1_0·b_1_2
- b_5_392 + b_1_24·b_1_3·b_5_40 + b_1_24·b_1_3·b_5_39 + b_1_25·b_5_39
+ b_1_1·b_1_23·b_1_3·b_5_40 + b_1_1·b_1_23·b_1_3·b_5_39 + b_1_1·b_1_24·b_5_40 + b_1_1·b_1_24·b_5_39 + b_1_12·b_1_22·b_1_3·b_5_40 + b_1_12·b_1_22·b_1_3·b_5_39 + b_1_12·b_1_23·b_5_40 + b_1_02·b_1_12·b_1_3·b_5_40 + b_1_02·b_1_12·b_1_3·b_5_39 + b_1_03·b_1_1·b_1_3·b_5_40 + b_1_03·b_1_1·b_1_2·b_5_40 + b_1_03·b_1_12·b_5_40 + b_1_03·b_1_12·b_5_39 + b_1_04·b_1_3·b_5_40 + b_1_06·b_1_12·b_1_3·b_1_4 + b_1_06·b_1_12·b_1_2·b_1_3 + b_1_07·b_1_1·b_1_3·b_1_4 + b_1_07·b_1_1·b_1_2·b_1_3 + b_1_08·b_1_3·b_1_4 + b_1_08·b_1_2·b_1_3 + b_1_08·b_1_1·b_1_4 + b_1_08·b_1_1·b_1_2 + c_8_87·b_1_22 + c_8_87·b_1_02 + c_4_31·b_1_24·b_1_3·b_1_4 + c_4_31·b_1_25·b_1_3 + c_4_31·b_1_26 + c_4_31·b_1_1·b_1_23·b_1_3·b_1_4 + c_4_31·b_1_12·b_1_23·b_1_4 + c_4_31·b_1_12·b_1_23·b_1_3 + c_4_31·b_1_02·b_1_12·b_1_2·b_1_3 + c_4_31·b_1_04·b_1_3·b_1_4 + c_4_31·b_1_04·b_1_1·b_1_4 + c_4_31·b_1_04·b_1_1·b_1_3 + c_4_31·b_1_04·b_1_12 + c_4_31·b_1_05·b_1_4 + c_4_31·b_1_05·b_1_1 + c_4_31·b_1_06 + c_4_312·b_1_22 + c_4_312·b_1_1·b_1_3 + c_4_312·b_1_12 + c_4_312·b_1_0·b_1_4 + c_4_312·b_1_0·b_1_2 + c_4_312·b_1_0·b_1_1
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_31, a Duflot regular element of degree 4
- c_8_87, a Duflot regular element of degree 8
- b_1_42 + b_1_22 + b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 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
- 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
- b_1_3 → 0, an element of degree 1
- b_1_4 → 0, an element of degree 1
- c_4_31 → c_1_14, an element of degree 4
- b_5_39 → 0, an element of degree 5
- b_5_40 → 0, an element of degree 5
- c_8_87 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_1_4 → 0, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_12·c_1_23 + 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_40 → c_1_12·c_1_23 + 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_87 → c_1_12·c_1_26 + c_1_18 + c_1_04·c_1_24 + 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_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_1_4 → 0, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_04·c_1_2, an element of degree 5
- b_5_40 → 0, an element of degree 5
- c_8_87 → c_1_12·c_1_26 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_1_4 → 0, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_25 + c_1_12·c_1_23 + 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_40 → c_1_25 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- c_8_87 → c_1_28 + c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_1_4 → 0, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- b_5_40 → c_1_12·c_1_23 + 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_87 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- c_4_31 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_40 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
- c_8_87 → c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24
+ c_1_08, an element of degree 8
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