Simon King
David J. Green
Cohomology
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Cohomology of group number 244 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 2.
- It has 2 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 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
( − 1) · (t2 − t + 1) · (t2 + t + 1) |
| (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 11 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_2_4, an element of degree 2
- a_3_5, a nilpotent element of degree 3
- a_3_6, a nilpotent element of degree 3
- b_4_6, an element of degree 4
- b_4_8, an element of degree 4
- c_4_9, a Duflot regular element of degree 4
- c_4_10, a Duflot regular element of degree 4
- a_5_15, a nilpotent element of degree 5
Ring relations
There are 35 minimal relations of maximal degree 10:
- a_1_02
- a_1_0·b_1_1
- a_1_0·a_1_22
- a_1_22·b_1_1
- b_2_4·b_1_1
- a_1_24
- b_1_1·a_3_5
- a_1_0·a_3_5
- b_1_1·a_3_6
- a_1_0·a_3_6 + b_2_4·a_1_22
- a_1_22·a_3_5
- a_1_22·a_3_6
- b_4_6·a_1_0 + b_2_4·a_1_23
- b_4_8·b_1_1
- b_4_8·a_1_0 + b_2_4·a_3_5 + b_2_4·a_1_23
- a_3_52
- a_3_62 + b_2_42·a_1_0·a_1_2
- b_2_4·b_4_6 + a_3_5·a_3_6 + b_2_4·a_1_2·a_3_6 + b_2_42·a_1_0·a_1_2
- b_4_6·a_1_22
- a_3_5·a_3_6 + b_4_8·a_1_22
- b_1_1·a_5_15 + b_4_6·a_1_2·b_1_1 + c_4_10·a_1_2·b_1_1
- a_3_5·a_3_6 + a_1_0·a_5_15 + b_2_4·a_1_2·a_3_5 + b_2_42·a_1_22 + b_2_42·a_1_0·a_1_2
+ c_4_10·a_1_0·a_1_2
- b_4_6·a_3_6 + b_2_42·a_1_23
- b_4_8·a_3_5 + b_4_6·a_3_5 + b_2_4·c_4_10·a_1_0
- b_4_6·a_3_5 + b_4_8·a_1_23
- b_4_8·a_3_6 + b_2_4·a_5_15 + b_2_4·b_4_8·a_1_2 + b_2_42·a_3_6 + b_2_43·a_1_2
+ b_2_4·c_4_10·a_1_2 + b_2_4·c_4_10·a_1_0 + b_2_4·c_4_9·a_1_0
- b_4_6·a_3_5 + a_1_22·a_5_15 + b_2_42·a_1_23 + c_4_10·a_1_23
- b_4_62 + c_4_10·b_1_14 + c_4_9·b_1_14
- b_4_82 + b_2_42·c_4_10
- b_4_6·b_4_8 + a_3_6·a_5_15 + b_2_42·a_1_2·a_3_6 + b_2_43·a_1_0·a_1_2
+ c_4_10·a_1_2·a_3_6 + b_2_4·c_4_9·a_1_22
- a_3_5·a_5_15 + b_2_4·b_4_8·a_1_22 + b_2_42·a_1_2·a_3_5 + c_4_10·a_1_2·a_3_5
+ b_2_4·c_4_10·a_1_22 + b_2_4·c_4_10·a_1_0·a_1_2
- b_4_6·b_4_8 + b_2_4·a_1_2·a_5_15 + b_2_4·b_4_8·a_1_22 + b_2_42·a_1_2·a_3_6
+ b_2_42·a_1_2·a_3_5 + b_2_43·a_1_22 + b_2_4·c_4_10·a_1_0·a_1_2 + b_2_4·c_4_9·a_1_0·a_1_2
- b_4_6·a_5_15 + b_2_43·a_1_23 + c_4_10·a_1_2·b_1_14 + c_4_9·a_1_2·b_1_14
+ b_4_6·c_4_10·a_1_2 + b_2_4·c_4_9·a_1_23
- b_4_8·a_5_15 + b_2_42·a_5_15 + b_2_43·a_3_6 + b_2_44·a_1_2 + b_4_8·c_4_10·a_1_2
+ b_2_4·c_4_10·a_3_6 + b_2_4·c_4_10·a_3_5 + b_2_4·c_4_9·a_3_5 + b_2_42·c_4_10·a_1_0 + b_2_42·c_4_9·a_1_0 + b_2_4·c_4_10·a_1_23 + b_2_4·c_4_9·a_1_23
- a_5_152 + b_2_44·a_1_22 + b_2_44·a_1_0·a_1_2 + b_2_42·c_4_10·a_1_22
+ b_2_42·c_4_10·a_1_0·a_1_2 + c_4_102·a_1_22
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_4_9, a Duflot regular element of degree 4
- c_4_10, a Duflot regular element of degree 4
- b_1_12 + b_2_4, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 7].
- 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
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_2_4 → 0, an element of degree 2
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- b_4_6 → 0, an element of degree 4
- b_4_8 → 0, an element of degree 4
- c_4_9 → c_1_04, an element of degree 4
- c_4_10 → c_1_14 + c_1_04, an element of degree 4
- a_5_15 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_2_4 → 0, an element of degree 2
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- b_4_6 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- b_4_8 → 0, an element of degree 4
- c_4_9 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_10 → c_1_1·c_1_23 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_15 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_2_4 → c_1_22, an element of degree 2
- a_3_5 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- b_4_6 → 0, an element of degree 4
- b_4_8 → c_1_12·c_1_22 + c_1_02·c_1_22, an element of degree 4
- c_4_9 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_10 → c_1_14 + c_1_04, an element of degree 4
- a_5_15 → 0, an element of degree 5
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