Simon King
David J. Green
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Cohomology of group number 1401 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 5.
- Its center has rank 3.
- It has 4 conjugacy classes of maximal elementary abelian subgroups, which are of rank 4, 4, 4 and 5, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 4.
- The depth exceeds the Duflot bound, which is 3.
- The Poincaré series is
t2 − t − 1 |
| (t + 1)2 · (t − 1)5 |
- The a-invariants are -∞,-∞,-∞,-∞,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 8 minimal generators of maximal degree 3:
- 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
- c_2_7, a Duflot regular element of degree 2
- c_2_8, a Duflot regular element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_3_20, an element of degree 3
Ring relations
There are 6 minimal relations of maximal degree 6:
- b_1_1·b_1_2
- b_1_0·b_1_3 + b_1_0·b_1_2
- b_1_1·b_1_3 + b_1_0·b_1_2 + b_1_0·b_1_1 + b_1_02
- b_1_0·b_3_20 + c_2_9·b_1_02 + c_2_8·b_1_0·b_1_2
- b_1_1·b_3_20 + c_2_9·b_1_0·b_1_1 + c_2_8·b_1_0·b_1_2 + c_2_8·b_1_0·b_1_1
+ c_2_8·b_1_02
- b_3_202 + b_1_2·b_1_32·b_3_20 + b_1_22·b_1_3·b_3_20 + c_2_9·b_1_22·b_1_32
+ c_2_9·b_1_24 + c_2_8·b_1_34 + c_2_8·b_1_2·b_1_33 + c_2_7·b_1_24 + c_2_7·b_1_03·b_1_1 + c_2_7·b_1_04 + c_2_92·b_1_02 + c_2_82·b_1_32
Data used for Benson′s test
- Benson′s completion test succeeded in degree 6.
- 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_2_7, a Duflot regular element of degree 2
- c_2_8, a Duflot regular element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_12, an element of degree 2
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 3, 5].
- 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 3
- 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
- c_2_7 → c_1_12, an element of degree 2
- c_2_8 → c_1_22, an element of degree 2
- c_2_9 → c_1_12 + c_1_02, an element of degree 2
- b_3_20 → 0, an element of degree 3
Restriction map to a maximal el. ab. subgp. of rank 4
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_3, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_7 → c_1_12, an element of degree 2
- c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- c_2_9 → c_1_12 + c_1_0·c_1_3 + c_1_02, an element of degree 2
- b_3_20 → 0, an element of degree 3
Restriction map to a maximal el. ab. subgp. of rank 4
- b_1_0 → c_1_3, an element of degree 1
- b_1_1 → c_1_3, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_7 → c_1_1·c_1_3 + c_1_12, an element of degree 2
- c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- c_2_9 → c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_3 + c_1_02, an element of degree 2
- b_3_20 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
Restriction map to a maximal el. ab. subgp. of rank 4
- b_1_0 → c_1_3, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- c_2_7 → c_1_1·c_1_3 + c_1_12, an element of degree 2
- c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- c_2_9 → c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_3 + c_1_02, an element of degree 2
- b_3_20 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32
+ c_1_02·c_1_3, an element of degree 3
Restriction map to a maximal el. ab. subgp. of rank 5
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → c_1_4, an element of degree 1
- c_2_7 → c_1_1·c_1_4 + c_1_1·c_1_3 + c_1_12, an element of degree 2
- c_2_8 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- c_2_9 → c_1_1·c_1_4 + c_1_1·c_1_3 + c_1_12 + c_1_0·c_1_4 + c_1_02, an element of degree 2
- b_3_20 → c_1_2·c_1_42 + c_1_22·c_1_4 + c_1_1·c_1_3·c_1_4 + c_1_0·c_1_3·c_1_4 + c_1_0·c_1_32, an element of degree 3
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