Simon King
David J. Green
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Cohomology of group number 2023 of order 128
General information on the group
- The group is also known as SD16*SD16, the Central product SD_16 * SD_16.
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 1.
- It has 4 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 2.
- The depth exceeds the Duflot bound, which is 1.
- The Poincaré series is
( − 1) · (t6 − t5 + t2 + t + 1) |
| (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 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
- b_1_3, an element of degree 1
- b_2_8, an element of degree 2
- b_6_29, an element of degree 6
- b_6_30, an element of degree 6
- b_7_36, an element of degree 7
- c_8_43, a Duflot regular element of degree 8
Ring relations
There are 18 minimal relations of maximal degree 14:
- b_1_0·b_1_2
- b_1_1·b_1_3 + b_1_0·b_1_1
- b_1_1·b_1_22 + b_1_02·b_1_3 + b_1_03 + b_2_8·b_1_3 + b_2_8·b_1_2 + b_2_8·b_1_1
- b_2_8·b_1_12·b_1_2 + b_2_8·b_1_0·b_1_32 + b_2_8·b_1_03
- b_6_30·b_1_2 + b_2_8·b_1_25 + b_2_82·b_1_23 + b_2_82·b_1_02·b_1_3
+ b_2_82·b_1_03 + b_2_83·b_1_2
- b_1_03·b_1_14 + b_1_04·b_1_13 + b_1_05·b_1_12 + b_6_30·b_1_3 + b_6_29·b_1_0
+ b_2_8·b_1_24·b_1_3 + b_2_8·b_1_04·b_1_1 + b_2_8·b_1_05 + b_2_82·b_1_22·b_1_3 + b_2_82·b_1_02·b_1_1 + b_2_82·b_1_03 + b_2_83·b_1_2 + b_2_83·b_1_1
- b_1_03·b_1_14 + b_1_04·b_1_13 + b_1_05·b_1_12 + b_6_30·b_1_0 + b_6_29·b_1_0
+ b_2_82·b_1_02·b_1_3 + b_2_82·b_1_02·b_1_1 + b_2_83·b_1_0
- b_1_16·b_1_2 + b_1_02·b_1_15 + b_1_03·b_1_14 + b_1_04·b_1_13 + b_6_30·b_1_1
+ b_6_29·b_1_1 + b_2_82·b_1_22·b_1_3 + b_2_82·b_1_23 + b_2_82·b_1_02·b_1_3 + b_2_82·b_1_03 + b_2_83·b_1_1
- b_1_2·b_7_36 + b_6_29·b_1_2·b_1_3 + b_6_29·b_1_22 + b_2_8·b_1_25·b_1_3
+ b_2_8·b_1_05·b_1_1 + b_2_8·b_6_30 + b_2_8·b_6_29 + b_2_82·b_1_24 + b_2_83·b_1_22 + b_2_83·b_1_12 + b_2_83·b_1_0·b_1_1 + b_2_84
- b_1_3·b_7_36 + b_1_05·b_1_13 + b_1_08 + b_6_30·b_1_0·b_1_1 + b_6_29·b_1_32
+ b_6_29·b_1_2·b_1_3 + b_2_8·b_1_36 + b_2_8·b_1_05·b_1_1 + b_2_8·b_1_06 + b_2_8·b_6_29 + b_2_82·b_1_03·b_1_1 + b_2_83·b_1_02
- b_1_0·b_7_36 + b_1_05·b_1_13 + b_1_08 + b_6_30·b_1_0·b_1_1 + b_6_29·b_1_02
+ b_2_8·b_6_30 + b_2_82·b_1_24 + b_2_82·b_1_04 + b_2_83·b_1_22 + b_2_83·b_1_12 + b_2_83·b_1_0·b_1_1 + b_2_83·b_1_02 + b_2_84
- b_1_1·b_7_36 + b_1_05·b_1_13 + b_1_06·b_1_12 + b_1_07·b_1_1 + b_6_30·b_1_12
+ b_6_30·b_1_02 + b_6_29·b_1_0·b_1_1 + b_6_29·b_1_02 + b_2_8·b_1_05·b_1_1 + b_2_8·b_1_06 + b_2_8·b_6_30 + b_2_82·b_1_24 + b_2_82·b_1_03·b_1_1 + b_2_83·b_1_22 + b_2_84
- b_6_30·b_1_02·b_1_14 + b_6_30·b_1_03·b_1_13 + b_6_30·b_1_04·b_1_12
+ b_6_302 + b_6_29·b_6_30 + b_2_8·b_6_29·b_1_24 + b_2_82·b_1_28 + b_2_82·b_6_29·b_1_22 + b_2_82·b_6_29·b_1_0·b_1_1 + b_2_82·b_6_29·b_1_02 + b_2_83·b_1_06 + b_2_83·b_6_29 + b_2_84·b_1_24 + b_2_86
- b_6_29·b_1_23·b_1_33 + b_6_29·b_1_02·b_1_14 + b_6_29·b_1_03·b_1_13
+ b_6_29·b_1_04·b_1_12 + b_6_29·b_6_30 + b_6_292 + b_2_8·b_1_110 + b_2_8·b_6_30·b_1_04 + b_2_8·b_6_29·b_1_24 + b_2_8·b_6_29·b_1_04 + b_2_82·b_6_29·b_1_22 + b_2_82·b_6_29·b_1_0·b_1_1 + b_2_82·b_6_29·b_1_02 + b_2_83·b_6_29 + b_2_84·b_1_04 + c_8_43·b_1_22·b_1_32
- b_1_010·b_1_12 + b_1_012 + b_6_30·b_1_02·b_1_14 + b_6_30·b_1_03·b_1_13
+ b_6_29·b_1_02·b_1_14 + b_6_29·b_1_03·b_1_13 + b_6_29·b_6_30 + b_2_8·b_6_29·b_1_24 + b_2_82·b_6_29·b_1_22 + b_2_83·b_1_06 + b_2_83·b_6_29 + c_8_43·b_1_02·b_1_12
- b_6_30·b_7_36 + b_6_30·b_1_04·b_1_13 + b_6_30·b_1_07 + b_6_302·b_1_1
+ b_6_29·b_7_36 + b_6_29·b_1_04·b_1_13 + b_6_29·b_1_07 + b_6_29·b_6_30·b_1_1 + b_6_29·b_6_30·b_1_0 + b_6_292·b_1_3 + b_6_292·b_1_2 + b_2_8·b_6_30·b_1_05 + b_2_82·b_1_28·b_1_3 + b_2_82·b_6_30·b_1_03 + b_2_82·b_6_29·b_1_23 + b_2_82·b_6_29·b_1_03 + b_2_83·b_7_36 + b_2_83·b_1_27 + b_2_83·b_6_29·b_1_2 + b_2_84·b_1_05 + b_2_85·b_1_02·b_1_1 + b_2_86·b_1_1 + b_2_8·c_8_43·b_1_22·b_1_3
- b_6_30·b_7_36 + b_6_30·b_1_04·b_1_13 + b_6_30·b_1_07 + b_6_302·b_1_1
+ b_6_29·b_6_30·b_1_0 + b_2_8·b_6_30·b_1_05 + b_2_82·b_1_28·b_1_3 + b_2_82·b_6_30·b_1_03 + b_2_82·b_6_29·b_1_23 + b_2_82·b_6_29·b_1_02·b_1_1 + b_2_82·b_6_29·b_1_03 + b_2_83·b_7_36 + b_2_83·b_1_27 + b_2_83·b_6_29·b_1_2 + b_2_83·b_6_29·b_1_0 + b_2_85·b_1_03 + b_2_86·b_1_1 + b_2_8·c_8_43·b_1_02·b_1_1
- b_7_362 + b_1_014 + b_6_30·b_1_05·b_1_13 + b_6_30·b_1_07·b_1_1 + b_6_30·b_1_08
+ b_6_302·b_1_02 + b_6_29·b_1_05·b_1_13 + b_6_29·b_1_07·b_1_1 + b_6_29·b_1_08 + b_6_29·b_6_30·b_1_12 + b_6_292·b_1_32 + b_6_292·b_1_22 + b_6_292·b_1_12 + b_6_292·b_1_02 + b_2_8·b_1_112 + b_2_8·b_6_302 + b_2_8·b_6_29·b_1_06 + b_2_82·b_1_29·b_1_3 + b_2_82·b_6_30·b_1_04 + b_2_82·b_6_29·b_1_24 + b_2_82·b_6_29·b_1_03·b_1_1 + b_2_82·b_6_29·b_1_04 + b_2_83·b_1_28 + b_2_83·b_6_29·b_1_02 + b_2_85·b_1_24 + b_2_86·b_1_02 + b_2_87 + c_8_43·b_1_02·b_1_14 + b_2_8·c_8_43·b_1_03·b_1_1 + b_2_82·c_8_43·b_1_2·b_1_3 + b_2_82·c_8_43·b_1_0·b_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 14.
- 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_43, a Duflot regular element of degree 8
- b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_2 + b_1_12 + b_1_0·b_1_1 + b_2_8, an element of degree 2
- b_1_2·b_1_32 + b_1_22·b_1_3 + b_1_12·b_1_2 + b_1_0·b_1_12 + b_1_02·b_1_3
+ b_1_02·b_1_1 + b_1_03 + b_2_8·b_1_2 + b_2_8·b_1_1, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, 6, 10].
- 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
- b_1_3 → 0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- b_6_29 → 0, an element of degree 6
- b_6_30 → 0, an element of degree 6
- b_7_36 → 0, an element of degree 7
- c_8_43 → c_1_08, an element of degree 8
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 + c_1_1, 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_2_8 → 0, an element of degree 2
- b_6_29 → c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_0·c_1_1·c_1_24
+ c_1_0·c_1_13·c_1_22 + c_1_02·c_1_24 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2, an element of degree 6
- b_6_30 → c_1_26 + c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_0·c_1_1·c_1_24
+ c_1_0·c_1_13·c_1_22 + c_1_02·c_1_24 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2, an element of degree 6
- b_7_36 → c_1_27 + c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_13·c_1_24 + c_1_14·c_1_23
+ c_1_15·c_1_22 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
- c_8_43 → c_1_15·c_1_23 + c_1_16·c_1_22 + c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25
+ c_1_0·c_1_14·c_1_23 + c_1_0·c_1_15·c_1_22 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_15·c_1_2 + 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_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- 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
- b_1_3 → c_1_2, an element of degree 1
- b_2_8 → 0, an element of degree 2
- b_6_29 → c_1_0·c_1_12·c_1_23 + c_1_0·c_1_13·c_1_22 + c_1_02·c_1_1·c_1_23
+ c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_1·c_1_2, an element of degree 6
- b_6_30 → 0, an element of degree 6
- b_7_36 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
+ c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
- c_8_43 → c_1_0·c_1_13·c_1_24 + c_1_0·c_1_14·c_1_23 + c_1_02·c_1_13·c_1_23
+ c_1_04·c_1_24 + 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 → 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
- b_1_3 → c_1_1, an element of degree 1
- b_2_8 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_6_29 → c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2
+ c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
- b_6_30 → c_1_26 + c_1_1·c_1_25 + c_1_13·c_1_23 + c_1_15·c_1_2, an element of degree 6
- b_7_36 → c_1_13·c_1_24 + c_1_16·c_1_2 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
+ c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
- c_8_43 → c_1_14·c_1_24 + c_1_17·c_1_2 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
+ c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8
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 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_2_8 → c_1_1·c_1_2 + c_1_12, an element of degree 2
- b_6_29 → c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23
+ c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22, an element of degree 6
- b_6_30 → c_1_26 + c_1_1·c_1_25 + c_1_12·c_1_24 + c_1_13·c_1_23 + c_1_14·c_1_22
+ c_1_15·c_1_2 + c_1_16 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_04·c_1_22, an element of degree 6
- b_7_36 → c_1_27 + c_1_1·c_1_26 + c_1_12·c_1_25 + c_1_0·c_1_12·c_1_24
+ c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
- c_8_43 → c_1_1·c_1_27 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
+ c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
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