Simon King
David J. Green
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Cohomology of group number 2224 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 2.
- It has 5 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 3, 3 and 4, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
( − 1) · (t3 − t2 − t − 1) |
| (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 8 minimal generators of maximal degree 4:
- 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
- b_4_31, an element of degree 4
- c_4_32, a Duflot regular element of degree 4
- c_4_33, a Duflot regular element of degree 4
Ring relations
There are 6 minimal relations of maximal degree 8:
- b_1_12 + b_1_0·b_1_2
- b_1_32 + b_1_1·b_1_3 + 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_0·b_1_42 + b_1_0·b_1_1·b_1_4 + b_1_02·b_1_4 + b_1_02·b_1_1
- b_1_02·b_1_2·b_1_3·b_1_4 + b_1_02·b_1_1·b_1_3·b_1_4 + b_1_02·b_1_1·b_1_2·b_1_3
+ b_1_03·b_1_1·b_1_3 + b_1_04·b_1_4 + b_1_04·b_1_2 + b_4_31·b_1_0
- b_4_31·b_1_44 + b_4_31·b_1_22·b_1_42 + b_4_31·b_1_24
+ b_4_31·b_1_1·b_1_22·b_1_4 + b_4_31·b_1_02·b_1_1·b_1_4 + b_4_31·b_1_03·b_1_1 + b_4_312 + c_4_33·b_1_24 + c_4_33·b_1_03·b_1_2 + c_4_32·b_1_44 + c_4_32·b_1_24 + c_4_32·b_1_0·b_1_1·b_1_2·b_1_4 + c_4_32·b_1_02·b_1_2·b_1_4 + c_4_32·b_1_02·b_1_1·b_1_4 + c_4_32·b_1_02·b_1_1·b_1_2 + c_4_32·b_1_03·b_1_4 + c_4_32·b_1_03·b_1_1
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_4_32, a Duflot regular element of degree 4
- c_4_33, a Duflot regular element of degree 4
- b_1_42 + b_1_3·b_1_4 + b_1_2·b_1_4 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_4 + b_1_02, an element of degree 2
- b_1_42, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 6, 8].
- 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 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
- b_4_31 → 0, an element of degree 4
- c_4_32 → c_1_14, an element of degree 4
- c_4_33 → c_1_14 + c_1_04, an element of degree 4
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
- b_4_31 → 0, an element of degree 4
- c_4_32 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- c_4_33 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
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 → c_1_2, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- b_4_31 → c_1_24, an element of degree 4
- c_4_32 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- c_4_33 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
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 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- b_4_31 → 0, an element of degree 4
- c_4_32 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- c_4_33 → c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
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 → 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
- b_4_31 → 0, an element of degree 4
- c_4_32 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- c_4_33 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- 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 → c_1_3, an element of degree 1
- b_4_31 → c_1_12·c_1_32 + c_1_02·c_1_22, an element of degree 4
- c_4_32 → c_1_12·c_1_32 + c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22, an element of degree 4
- c_4_33 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
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