Simon King
David J. Green
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Cohomology of group number 2288 of order 128
General information on the group
- The group has 5 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t12 + 3·t11 + 2·t10 + t9 − 2·t8 − 2·t7 + 2·t5 − t4 − 2·t3 − 3·t2 − 2·t − 1 |
| (t − 1)3 · (t2 + 1)2 · (t4 + 1)2 |
- The a-invariants are -∞,-∞,-3,-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 8:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_4, an element of degree 1
- a_6_26, a nilpotent element of degree 6
- a_6_27, a nilpotent element of degree 6
- b_6_28, an element of degree 6
- b_6_29, an element of degree 6
- c_8_36, a Duflot regular element of degree 8
- c_8_37, a Duflot regular element of degree 8
Ring relations
There are 22 minimal relations of maximal degree 12:
- a_1_1·b_1_4 + a_1_32 + a_1_22 + a_1_1·a_1_3 + a_1_1·a_1_2 + a_1_0·a_1_3 + a_1_0·a_1_2
+ a_1_0·a_1_1 + a_1_02
- a_1_0·b_1_4 + a_1_22 + a_1_1·a_1_2 + a_1_12 + a_1_0·a_1_3 + a_1_0·a_1_2
- a_1_32·b_1_4 + a_1_33 + a_1_22·a_1_3 + a_1_1·a_1_2·a_1_3 + a_1_12·a_1_3
+ a_1_0·a_1_2·a_1_3 + a_1_0·a_1_22 + a_1_0·a_1_1·a_1_2 + a_1_0·a_1_12 + a_1_02·a_1_2 + a_1_03
- a_1_22·b_1_4 + a_1_2·a_1_32 + a_1_22·a_1_3 + a_1_1·a_1_32 + a_1_12·a_1_2
+ a_1_0·a_1_2·a_1_3 + a_1_0·a_1_1·a_1_3 + a_1_0·a_1_1·a_1_2 + a_1_02·a_1_2 + a_1_02·a_1_1
- a_1_02·a_1_13 + a_1_03·a_1_12 + a_1_05
- a_1_02·a_1_13 + a_1_04·a_1_1 + a_1_05
- b_6_28·a_1_1 + a_6_27·a_1_0 + a_6_26·a_1_1 + a_1_0·a_1_12·a_1_2·a_1_33
+ a_1_02·a_1_1·a_1_2·a_1_33 + a_1_03·a_1_2·a_1_33 + a_1_04·a_1_33 + a_1_04·a_1_2·a_1_32 + a_1_05·a_1_2·a_1_3
- b_6_28·a_1_0 + a_6_26·a_1_1 + a_6_26·a_1_0 + a_1_03·a_1_23·a_1_3 + a_1_04·a_1_33
+ a_1_04·a_1_2·a_1_32 + a_1_05·a_1_2·a_1_3
- b_6_29·a_1_1 + a_6_27·a_1_1 + a_6_26·a_1_1 + a_6_26·a_1_0 + a_1_0·a_1_12·a_1_2·a_1_33
+ a_1_02·a_1_1·a_1_2·a_1_33 + a_1_04·a_1_33 + a_1_04·a_1_22·a_1_3 + a_1_04·a_1_23 + a_1_05·a_1_2·a_1_3
- b_6_29·a_1_0 + a_6_27·a_1_1 + a_6_27·a_1_0 + a_6_26·a_1_1 + a_6_26·a_1_0
+ a_1_02·a_1_1·a_1_2·a_1_33 + a_1_04·a_1_33 + a_1_04·a_1_2·a_1_32 + a_1_04·a_1_22·a_1_3
- a_6_26·b_1_42 + a_6_27·a_1_0·a_1_1 + a_6_26·a_1_22 + a_6_26·a_1_1·a_1_2
+ a_6_26·a_1_12 + a_6_26·a_1_0·a_1_3 + a_6_26·a_1_0·a_1_2 + a_6_26·a_1_0·a_1_1 + a_6_26·a_1_02 + a_1_04·a_1_2·a_1_33
- a_6_27·b_1_42 + a_6_27·a_1_22 + a_6_27·a_1_1·a_1_2 + a_6_27·a_1_0·a_1_3
+ a_6_27·a_1_0·a_1_2 + a_6_26·a_1_0·a_1_1 + a_6_26·a_1_02 + a_1_04·a_1_2·a_1_33 + a_1_04·a_1_23·a_1_3
- a_6_26·a_1_0·a_1_12 + a_6_26·a_1_02·a_1_1
- a_6_272 + a_6_26·a_6_27 + a_6_262 + a_6_26·a_1_12·a_1_2·a_1_33
+ a_6_26·a_1_02·a_1_23·a_1_3 + a_6_26·a_1_03·a_1_22·a_1_3 + a_6_26·a_1_03·a_1_23
- a_6_27·b_6_29 + a_6_26·b_6_28 + a_6_26·a_1_02·a_1_2·a_1_33
+ a_6_26·a_1_02·a_1_23·a_1_3 + a_6_26·a_1_03·a_1_2·a_1_32 + a_6_26·a_1_03·a_1_22·a_1_3 + a_6_26·a_1_03·a_1_23
- a_6_27·b_6_28 + a_6_26·b_6_29 + a_6_26·b_6_28 + a_6_26·a_1_02·a_1_2·a_1_33
+ a_6_26·a_1_03·a_1_33 + a_6_26·a_1_03·a_1_2·a_1_32 + a_6_26·a_1_03·a_1_22·a_1_3 + a_6_26·a_1_03·a_1_23 + a_6_26·a_1_03·a_1_1·a_1_2·a_1_3 + a_6_26·a_1_04·a_1_2·a_1_3
- b_6_28·b_1_46 + b_6_282 + b_6_28·a_1_3·b_1_45 + b_6_28·a_1_2·b_1_45
+ b_6_29·a_1_2·a_1_3·b_1_44 + a_6_272 + a_6_26·a_1_12·a_1_2·a_1_33 + a_6_26·a_1_02·a_1_2·a_1_33 + a_6_26·a_1_03·a_1_22·a_1_3 + a_6_26·a_1_04·a_1_2·a_1_3 + c_8_36·b_1_44 + c_8_36·a_1_0·a_1_13 + c_8_36·a_1_02·a_1_22 + c_8_36·a_1_02·a_1_1·a_1_2 + c_8_36·a_1_03·a_1_3 + c_8_36·a_1_03·a_1_2 + c_8_36·a_1_03·a_1_1 + c_8_36·a_1_04
- a_6_27·b_6_28 + a_6_272 + a_6_262 + a_6_26·a_1_12·a_1_2·a_1_33
+ a_6_26·a_1_03·a_1_2·a_1_32 + a_6_26·a_1_03·a_1_22·a_1_3 + a_6_26·a_1_03·a_1_1·a_1_2·a_1_3 + a_6_26·a_1_04·a_1_2·a_1_3 + c_8_37·a_1_0·a_1_13 + c_8_36·a_1_03·a_1_1
- a_6_272 + a_6_262 + a_6_26·a_1_03·a_1_33 + c_8_37·a_1_02·a_1_12
+ c_8_36·a_1_04
- a_6_26·b_6_28 + a_6_262 + a_6_26·a_1_02·a_1_23·a_1_3 + a_6_26·a_1_03·a_1_33
+ a_6_26·a_1_03·a_1_2·a_1_32 + a_6_26·a_1_03·a_1_22·a_1_3 + a_6_26·a_1_04·a_1_2·a_1_3 + c_8_37·a_1_03·a_1_1 + c_8_36·a_1_0·a_1_13 + c_8_36·a_1_03·a_1_1
- a_6_262 + a_6_26·a_1_03·a_1_33 + a_6_26·a_1_03·a_1_1·a_1_2·a_1_3
+ c_8_37·a_1_04 + c_8_36·a_1_02·a_1_12 + c_8_36·a_1_04
- b_6_29·b_1_46 + b_6_292 + b_6_28·b_1_46 + b_6_29·a_1_2·b_1_45 + a_6_27·b_6_28
+ a_6_26·b_6_28 + a_6_272 + a_6_26·a_1_12·a_1_2·a_1_33 + a_6_26·a_1_02·a_1_23·a_1_3 + a_6_26·a_1_03·a_1_2·a_1_32 + a_6_26·a_1_03·a_1_23 + a_6_26·a_1_03·a_1_1·a_1_2·a_1_3 + a_6_26·a_1_04·a_1_2·a_1_3 + c_8_37·b_1_44 + c_8_37·a_1_02·a_1_22 + c_8_37·a_1_02·a_1_1·a_1_2 + c_8_37·a_1_03·a_1_3 + c_8_37·a_1_03·a_1_2 + c_8_36·a_1_0·a_1_13 + c_8_36·a_1_02·a_1_12 + c_8_36·a_1_04
Data used for Benson′s test
- Benson′s completion test succeeded in degree 15.
- However, the last relation was already found in degree 12 and the last generator in degree 8.
- The following is a filter regular homogeneous system of parameters:
- c_8_36, a Duflot regular element of degree 8
- c_8_37, a Duflot regular element of degree 8
- b_1_42, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 13, 15].
- 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_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_4 → 0, an element of degree 1
- a_6_26 → 0, an element of degree 6
- a_6_27 → 0, an element of degree 6
- b_6_28 → 0, an element of degree 6
- b_6_29 → 0, an element of degree 6
- c_8_36 → c_1_18, an element of degree 8
- c_8_37 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_4 → c_1_2, an element of degree 1
- a_6_26 → 0, an element of degree 6
- a_6_27 → 0, an element of degree 6
- b_6_28 → c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
- b_6_29 → c_1_12·c_1_24 + c_1_14·c_1_22 + c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6
- c_8_36 → c_1_12·c_1_26 + c_1_18, an element of degree 8
- c_8_37 → c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_08, an element of degree 8
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