Simon King
David J. Green
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Singular
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Cohomology of group number 331 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 4.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t5 − t3 + t2 + 1 |
| (t + 1) · (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-4,-4,-4. 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_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_2_4, an element of degree 2
- b_2_5, an element of degree 2
- b_3_7, an element of degree 3
- b_3_8, an element of degree 3
- b_4_10, an element of degree 4
- c_4_11, a Duflot regular element of degree 4
- c_4_12, a Duflot regular element of degree 4
- b_5_19, an element of degree 5
Ring relations
There are 25 minimal relations of maximal degree 10:
- a_1_1·b_1_0
- a_1_2·b_1_0
- a_1_13
- b_2_4·a_1_1 + a_1_23 + a_1_12·a_1_2
- b_2_4·a_1_2 + a_1_1·a_1_22 + a_1_12·a_1_2
- b_2_5·a_1_1 + a_1_23 + a_1_1·a_1_22 + a_1_12·a_1_2
- a_1_1·b_3_7
- a_1_2·b_3_7 + a_1_12·a_1_22
- a_1_1·b_3_8 + a_1_1·a_1_23
- b_1_0·b_3_8 + b_2_42 + b_2_5·a_1_22
- a_1_2·b_3_8 + b_2_5·a_1_22 + a_1_1·a_1_23
- b_4_10·a_1_1
- b_4_10·b_1_0 + b_2_5·b_3_7 + b_2_4·b_3_7 + b_2_4·b_1_03
- b_3_82 + b_2_42·b_2_5 + b_2_43 + b_2_52·a_1_22 + c_4_11·b_1_02
- b_3_72 + c_4_12·b_1_02
- a_1_1·b_5_19 + c_4_12·a_1_1·a_1_2 + c_4_11·a_1_12
- b_3_82 + b_3_7·b_3_8 + b_1_0·b_5_19 + b_2_4·b_1_0·b_3_7 + b_2_4·b_1_04
+ b_2_4·b_2_5·b_1_02 + b_2_42·b_2_5 + b_2_43 + b_2_52·a_1_22
- a_1_2·b_5_19 + b_4_10·a_1_22 + b_2_52·a_1_22 + c_4_12·a_1_22 + c_4_11·a_1_1·a_1_2
- b_4_10·b_3_7 + b_2_4·b_1_02·b_3_7 + b_2_5·c_4_12·b_1_0 + b_2_4·c_4_12·b_1_0
- b_4_10·b_3_8 + b_2_5·b_5_19 + b_2_4·b_5_19 + b_2_4·b_2_5·b_3_7 + b_2_4·b_2_5·b_1_03
+ b_2_4·b_2_52·b_1_0 + b_2_42·b_3_7 + b_2_42·b_1_03 + b_2_42·b_2_5·b_1_0 + b_2_43·b_1_0 + b_2_53·a_1_2 + b_2_5·c_4_11·b_1_0 + b_2_4·c_4_11·b_1_0 + b_2_5·c_4_12·a_1_2 + c_4_12·a_1_1·a_1_22 + c_4_12·a_1_12·a_1_2 + c_4_11·a_1_1·a_1_22
- b_4_102 + b_2_42·b_1_04 + b_2_52·c_4_12 + b_2_42·c_4_12
- b_3_7·b_5_19 + b_2_4·b_1_03·b_3_7 + b_2_4·b_2_5·b_1_0·b_3_7 + c_4_11·b_1_0·b_3_7
+ b_2_4·c_4_12·b_1_02 + b_2_42·c_4_12 + b_2_5·c_4_12·a_1_22 + c_4_12·a_1_12·a_1_22
- b_3_8·b_5_19 + b_2_4·b_1_0·b_5_19 + b_2_42·b_1_0·b_3_7 + b_2_42·b_1_04
+ b_2_42·b_4_10 + b_2_43·b_1_02 + b_2_43·b_2_5 + b_2_44 + b_2_5·b_4_10·a_1_22 + b_2_53·a_1_22 + c_4_11·b_1_0·b_3_7 + c_4_11·b_1_04 + b_2_4·c_4_11·b_1_02 + b_2_42·c_4_11 + b_2_5·c_4_12·a_1_22 + b_2_5·c_4_11·a_1_22 + c_4_12·a_1_1·a_1_23 + c_4_11·a_1_1·a_1_23
- b_4_10·b_5_19 + b_2_4·b_2_5·b_1_02·b_3_7 + b_2_4·b_2_52·b_3_7 + b_2_42·b_1_05
+ b_2_42·b_2_5·b_3_7 + b_2_43·b_3_7 + b_2_43·b_1_03 + b_2_44·b_1_0 + b_2_52·b_4_10·a_1_2 + c_4_11·b_1_05 + b_2_5·c_4_12·b_3_8 + b_2_5·c_4_11·b_3_7 + b_2_4·c_4_12·b_3_8 + b_2_4·c_4_11·b_3_7 + b_2_4·c_4_11·b_1_03 + b_2_4·b_2_5·c_4_12·b_1_0 + b_2_42·c_4_12·b_1_0 + b_4_10·c_4_12·a_1_2
- b_5_192 + b_2_42·b_1_06 + b_2_44·b_1_02 + b_2_44·b_2_5 + b_2_45
+ b_2_54·a_1_22 + b_2_5·c_4_11·b_1_04 + b_2_4·c_4_11·b_1_04 + b_2_42·c_4_12·b_1_02 + b_2_42·b_2_5·c_4_12 + b_2_43·c_4_12 + b_2_52·c_4_12·a_1_22 + c_4_11·c_4_12·b_1_02 + c_4_112·b_1_02 + c_4_122·a_1_22 + c_4_112·a_1_12
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_11, a Duflot regular element of degree 4
- c_4_12, a Duflot regular element of degree 4
- 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, 4, 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
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_3_7 → 0, an element of degree 3
- b_3_8 → 0, an element of degree 3
- b_4_10 → 0, an element of degree 4
- c_4_11 → c_1_14, an element of degree 4
- c_4_12 → c_1_04, an element of degree 4
- b_5_19 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_2_4 → c_1_1·c_1_2, an element of degree 2
- b_2_5 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- b_3_7 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_3_8 → c_1_12·c_1_2, an element of degree 3
- b_4_10 → c_1_1·c_1_23 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_1·c_1_22
+ c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_1·c_1_2, an element of degree 4
- c_4_11 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_13·c_1_2 + c_1_14, an element of degree 4
- c_4_12 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_5_19 → c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_3 + c_1_1·c_1_24 + c_1_12·c_1_2·c_1_32
+ c_1_12·c_1_22·c_1_3 + c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_1·c_1_23 + c_1_0·c_1_12·c_1_22 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2, an element of degree 5
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