Simon King
David J. Green
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Cohomology of group number 925 of order 128
General information on the group
- The group has 3 minimal generators and exponent 16.
- 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, 2 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
t6 − t2 − 1 |
| (t − 1)3 · (t2 + 1) · (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 8 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
- b_2_5, an element of degree 2
- b_3_9, an element of degree 3
- b_5_16, an element of degree 5
- c_8_25, a Duflot regular element of degree 8
Ring relations
There are 14 minimal relations of maximal degree 10:
- b_1_0·b_1_1
- b_1_0·b_1_2
- a_2_4·b_1_0
- b_1_1·b_1_22 + b_2_5·b_1_2 + b_2_5·b_1_1
- b_2_5·b_1_12 + a_2_4·b_1_1·b_1_2 + a_2_42
- b_1_2·b_3_9 + b_2_5·b_1_22 + a_2_4·b_1_22 + a_2_4·b_2_5
- b_1_1·b_3_9 + b_2_5·b_1_1·b_1_2 + a_2_4·b_2_5
- a_2_4·b_2_5·b_1_2 + a_2_4·b_2_5·b_1_1 + a_2_42·b_1_2
- b_2_52·b_1_1 + a_2_4·b_3_9
- b_1_2·b_5_16 + b_2_5·b_1_24 + a_2_4·b_1_24 + a_2_42·b_1_22
- b_3_92 + b_1_0·b_5_16 + b_2_5·b_1_0·b_3_9 + b_2_5·b_1_04 + b_2_52·b_1_22 + b_2_53
+ a_2_4·b_2_52 + a_2_42·b_1_22 + a_2_43
- b_1_1·b_5_16 + b_2_52·b_1_22 + a_2_42·b_1_22 + a_2_42·b_2_5
- a_2_4·b_5_16
- b_5_162 + b_1_02·b_3_9·b_5_16 + b_1_05·b_5_16 + b_2_5·b_1_03·b_5_16
+ b_2_5·b_1_05·b_3_9 + b_2_52·b_1_26 + b_2_52·b_1_03·b_3_9 + b_2_52·b_1_06 + b_2_53·b_1_04 + b_2_54·b_1_02 + a_2_42·b_1_26 + c_8_25·b_1_02
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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_8_25, a Duflot regular element of degree 8
- b_1_22 + b_1_12 + b_1_02 + b_2_5, an element of degree 2
- b_1_02, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 9].
- 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 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
- b_2_5 → 0, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_5_16 → 0, an element of degree 5
- c_8_25 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- 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
- b_2_5 → 0, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_5_16 → 0, an element of degree 5
- c_8_25 → c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- 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
- b_2_5 → 0, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_5_16 → 0, an element of degree 5
- c_8_25 → 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 → 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
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_3_9 → c_1_23 + c_1_12·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
- b_5_16 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2
+ c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- c_8_25 → c_1_28 + c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24
+ c_1_17·c_1_2 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24 + c_1_0·c_1_14·c_1_23 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16 + c_1_03·c_1_15 + c_1_04·c_1_24 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12 + c_1_08, an element of degree 8
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