Simon King
David J. Green
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Cohomology of group number 1525 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 3.
- 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 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 2) · (t8 + 1/2·t7 − 1/2·t6 + 3/2·t5 − 3/2·t4 − t3 − 1/2·t2 − t − 1/2) |
| (t + 1)2 · (t − 1)4 · (t2 + 1)3 |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 4:
- a_1_0, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_1_3, an element of degree 1
- a_3_4, a nilpotent element of degree 3
- a_3_6, a nilpotent element of degree 3
- b_3_7, an element of degree 3
- b_3_8, an element of degree 3
- b_3_9, an element of degree 3
- c_4_15, a Duflot regular element of degree 4
- c_4_16, a Duflot regular element of degree 4
- c_4_17, a Duflot regular element of degree 4
Ring relations
There are 24 minimal relations of maximal degree 7:
- a_1_1·b_1_3 + a_1_0·a_1_1 + a_1_02
- a_1_0·b_1_3 + a_1_12 + a_1_22 + a_1_0·a_1_2
- a_1_2·b_1_3 + a_1_2·a_1_1 + a_1_02
- a_1_02·a_1_2 + a_1_03
- a_1_0·a_1_12 + a_1_0·a_1_22 + a_1_02·a_1_1
- a_1_22·a_1_1 + a_1_0·a_1_12 + a_1_0·a_1_2·a_1_1 + a_1_0·a_1_22
- b_1_3·a_3_6 + a_1_2·b_3_7 + a_1_1·a_3_6 + a_1_0·a_3_4
- b_1_3·a_3_6 + b_1_3·a_3_4 + a_1_1·b_3_8 + a_1_0·b_3_7 + a_1_1·a_3_4 + a_1_2·a_3_6
+ a_1_0·a_3_6
- b_1_3·a_3_4 + a_1_0·b_3_8 + a_1_0·b_3_7 + a_1_1·a_3_4 + a_1_0·a_3_6
- b_1_3·a_3_6 + a_1_2·b_3_8 + a_1_1·a_3_6 + a_1_2·a_3_6
- b_1_3·a_3_4 + a_1_0·b_3_9 + a_1_0·b_3_7 + a_1_1·a_3_6 + a_1_2·a_3_6 + a_1_2·a_3_4
+ a_1_0·a_3_4
- b_1_3·a_3_6 + a_1_1·b_3_9 + a_1_2·b_3_9 + a_1_0·b_3_7 + a_1_0·a_3_6
- a_1_12·a_3_4 + a_1_22·a_3_4 + a_1_0·a_1_1·a_3_4 + a_1_0·a_1_2·a_3_4 + a_1_02·a_3_4
- a_1_2·a_1_1·a_3_6 + a_1_22·a_3_6 + a_1_0·a_1_1·a_3_6 + a_1_0·a_1_2·a_3_6
- a_1_22·b_3_9 + a_1_2·a_1_1·a_3_4 + a_1_0·a_1_1·a_3_6 + a_1_0·a_1_2·a_3_6
+ a_1_02·a_3_6
- b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_8 + b_1_33·b_3_7 + a_3_62 + a_3_42
+ a_1_02·a_1_1·a_3_6 + c_4_15·b_1_32 + c_4_16·a_1_12 + c_4_15·a_1_02
- b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_8 + b_1_33·b_3_7 + a_3_6·b_3_8 + a_3_62
+ a_1_02·a_1_1·a_3_6 + a_1_02·a_1_1·a_3_4 + c_4_15·b_1_32 + c_4_16·a_1_2·a_1_1 + c_4_16·a_1_0·a_1_1 + c_4_15·a_1_12 + c_4_15·a_1_2·a_1_1 + c_4_15·a_1_0·a_1_1
- b_3_92 + b_1_33·b_3_8 + b_1_33·b_3_7 + a_3_62 + a_3_42 + a_1_02·a_1_1·a_3_6
+ c_4_17·b_1_32 + c_4_15·b_1_32 + c_4_15·a_1_12 + c_4_15·a_1_22
- b_3_92 + b_3_82 + b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_7 + b_1_36 + a_3_62
+ a_1_02·a_1_1·a_3_6 + c_4_16·b_1_32 + c_4_17·a_1_12 + c_4_16·a_1_22 + c_4_16·a_1_02 + c_4_15·a_1_12
- b_3_92 + b_3_82 + b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_7 + b_1_36 + a_3_4·b_3_8
+ a_3_4·b_3_7 + a_3_4·a_3_6 + a_1_02·a_1_1·a_3_4 + c_4_16·b_1_32 + c_4_17·a_1_0·a_1_1 + c_4_16·a_1_0·a_1_1 + c_4_15·a_1_12 + c_4_15·a_1_0·a_1_1 + c_4_15·a_1_0·a_1_2 + c_4_15·a_1_02
- b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_8 + b_1_33·b_3_7 + a_1_02·a_1_1·a_3_6
+ a_1_02·a_1_1·a_3_4 + c_4_15·b_1_32 + c_4_17·a_1_02 + c_4_16·a_1_22 + c_4_15·a_1_12 + c_4_15·a_1_22 + c_4_15·a_1_02
- b_3_92 + b_3_82 + b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_7 + b_1_36 + a_3_6·b_3_9
+ a_3_6·b_3_8 + a_3_4·b_3_9 + a_3_4·b_3_7 + a_3_62 + a_3_4·a_3_6 + a_3_42 + a_1_02·a_1_1·a_3_6 + c_4_16·b_1_32 + c_4_17·a_1_0·a_1_2 + c_4_16·a_1_22 + c_4_15·a_1_12 + c_4_15·a_1_22 + c_4_15·a_1_0·a_1_2 + c_4_15·a_1_02
- b_3_72 + b_1_33·b_3_9 + b_1_33·b_3_8 + b_1_33·b_3_7 + a_3_42 + a_1_02·a_1_1·a_3_6
+ a_1_02·a_1_1·a_3_4 + c_4_15·b_1_32 + c_4_17·a_1_22 + c_4_16·a_1_02 + c_4_15·a_1_12 + c_4_15·a_1_22
- a_1_2·a_3_4·b_3_9 + a_1_0·a_3_6·b_3_7 + a_1_1·a_3_4·a_3_6 + a_1_2·a_3_4·a_3_6
+ c_4_17·a_1_0·a_1_22 + c_4_17·a_1_02·a_1_1 + c_4_16·a_1_0·a_1_2·a_1_1 + c_4_16·a_1_03 + c_4_15·a_1_02·a_1_1
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- However, the last relation was already found in degree 7 and the last generator in degree 4.
- The following is a filter regular homogeneous system of parameters:
- c_4_15, a Duflot regular element of degree 4
- c_4_16, a Duflot regular element of degree 4
- c_4_17, a Duflot regular element of degree 4
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 10].
- 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 3
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_4 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- b_3_7 → 0, an element of degree 3
- b_3_8 → 0, an element of degree 3
- b_3_9 → 0, an element of degree 3
- c_4_15 → c_1_24, an element of degree 4
- c_4_16 → c_1_24 + c_1_14, an element of degree 4
- c_4_17 → c_1_24 + c_1_14 + c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- a_3_4 → 0, an element of degree 3
- a_3_6 → 0, an element of degree 3
- b_3_7 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_8 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- b_3_9 → c_1_1·c_1_32 + c_1_12·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_15 → c_1_34 + c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32, an element of degree 4
- c_4_16 → c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_14 + c_1_0·c_1_33 + c_1_02·c_1_32, an element of degree 4
- c_4_17 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_14 + c_1_0·c_1_33
+ c_1_04, an element of degree 4
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