Simon King
David J. Green
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Cohomology of group number 1868 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 2.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.
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
t4 − t3 + t + 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 10 minimal generators of maximal degree 6:
- a_1_1, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- b_3_10, an element of degree 3
- b_4_12, an element of degree 4
- b_4_13, an element of degree 4
- c_4_14, a Duflot regular element of degree 4
- c_4_15, a Duflot regular element of degree 4
- b_6_37, an element of degree 6
Ring relations
There are 21 minimal relations of maximal degree 12:
- a_1_1·b_1_0 + a_1_12
- b_1_22 + b_1_0·b_1_2 + a_1_12
- a_1_13
- b_1_2·b_1_32 + a_1_1·b_1_32 + a_1_12·b_1_3
- a_1_1·b_3_10
- b_1_2·b_1_3·b_3_10 + b_1_0·b_1_2·b_3_10 + b_1_03·b_1_2·b_1_3 + b_1_04·b_1_2
+ b_4_12·b_1_2 + b_4_12·a_1_1
- b_1_03·b_1_2·b_1_3 + b_1_04·b_1_2 + b_4_13·b_1_2 + b_4_12·b_1_2
- b_1_2·b_1_3·b_3_10 + b_1_0·b_1_2·b_3_10 + b_1_03·b_1_2·b_1_3 + b_1_04·b_1_2
+ b_4_12·b_1_2 + b_4_13·a_1_1
- b_1_32·b_3_10 + b_1_0·b_1_2·b_3_10 + b_1_02·b_3_10 + b_1_03·b_1_2·b_1_3
+ b_1_04·b_1_2 + b_4_13·b_1_0 + b_4_12·b_1_2
- b_3_102 + b_4_13·b_1_2·b_1_3 + b_4_12·b_1_2·b_1_3 + b_4_12·b_1_02 + c_4_14·b_1_02
+ c_4_14·a_1_12
- b_4_13·b_3_10 + b_4_13·b_1_0·b_1_2·b_1_3 + b_4_12·b_1_0·b_1_32 + b_4_12·b_1_03
+ b_4_12·a_1_1·b_1_32 + c_4_14·b_1_0·b_1_32 + c_4_14·b_1_0·b_1_2·b_1_3 + c_4_14·b_1_03 + c_4_14·a_1_1·b_1_32 + c_4_14·a_1_1·b_1_2·b_1_3
- b_6_37·b_1_2 + b_4_13·b_1_02·b_1_2 + b_4_12·b_1_0·b_1_2·b_1_3 + b_4_12·b_1_02·b_1_2
+ c_4_14·b_1_0·b_1_2·b_1_3 + c_4_14·b_1_02·b_1_2 + c_4_14·a_1_1·b_1_2·b_1_3 + c_4_15·a_1_12·b_1_3 + c_4_15·a_1_12·b_1_2
- b_6_37·a_1_1 + c_4_15·a_1_1·b_1_2·b_1_3 + c_4_15·a_1_12·b_1_3 + c_4_15·a_1_12·b_1_2
+ c_4_14·a_1_12·b_1_2
- b_1_03·b_1_3·b_3_10 + b_1_04·b_3_10 + b_6_37·b_1_0 + b_4_13·b_1_0·b_1_32
+ b_4_13·b_1_02·b_1_3 + b_4_13·b_1_02·b_1_2 + b_4_12·b_3_10 + b_4_12·b_1_0·b_1_2·b_1_3 + b_4_12·b_1_02·b_1_2 + b_4_12·b_1_03 + b_4_12·a_1_1·b_1_32 + c_4_15·b_1_0·b_1_2·b_1_3 + c_4_15·b_1_02·b_1_3 + c_4_15·b_1_02·b_1_2 + c_4_15·b_1_03 + c_4_14·b_1_0·b_1_32 + c_4_14·b_1_0·b_1_2·b_1_3 + c_4_14·b_1_02·b_1_3 + c_4_14·a_1_1·b_1_32 + c_4_14·a_1_1·b_1_2·b_1_3 + c_4_14·a_1_12·b_1_3 + c_4_14·a_1_12·b_1_2
- b_4_13·b_1_03·b_1_2 + b_4_132 + b_4_12·b_1_34 + b_4_12·b_1_2·b_3_10
+ b_4_12·b_1_02·b_1_2·b_1_3 + b_4_12·b_1_03·b_1_2 + b_4_12·b_1_04 + c_4_14·b_1_34 + c_4_14·b_1_02·b_1_2·b_1_3 + c_4_14·b_1_03·b_1_2 + c_4_14·b_1_04 + c_4_14·a_1_12·b_1_2·b_1_3
- b_1_05·b_3_10 + b_6_37·b_1_32 + b_4_13·b_1_34 + b_4_13·b_1_0·b_1_33
+ b_4_12·b_1_34 + b_4_12·b_1_0·b_3_10 + b_4_12·b_1_03·b_1_3 + b_4_12·b_4_13 + b_4_122 + b_4_12·a_1_1·b_1_33 + c_4_15·b_1_0·b_1_33 + c_4_15·b_1_02·b_1_32 + c_4_15·b_1_03·b_1_2 + c_4_15·b_1_04 + c_4_14·b_1_0·b_1_33 + c_4_14·b_1_02·b_1_2·b_1_3 + c_4_14·b_1_04 + c_4_15·a_1_1·b_1_33 + c_4_14·a_1_1·b_1_33 + c_4_15·a_1_12·b_1_2·b_1_3
- b_1_05·b_3_10 + b_6_37·b_1_02 + b_4_13·b_1_02·b_1_32 + b_4_13·b_1_03·b_1_2
+ b_4_13·b_1_04 + b_4_12·b_1_34 + b_4_12·b_1_0·b_3_10 + b_4_12·b_1_02·b_1_32 + b_4_12·b_1_03·b_1_3 + b_4_12·b_1_03·b_1_2 + b_4_12·b_1_04 + b_4_122 + c_4_15·b_1_02·b_1_2·b_1_3 + c_4_15·b_1_03·b_1_3 + c_4_14·b_1_34 + c_4_14·b_1_02·b_1_32 + c_4_14·b_1_02·b_1_2·b_1_3 + c_4_14·b_1_03·b_1_3 + c_4_14·b_1_04
- b_6_37·b_3_10 + b_4_12·b_1_0·b_1_34 + b_4_12·b_1_02·b_3_10
+ b_4_12·b_1_02·b_1_33 + b_4_12·b_1_03·b_1_32 + b_4_12·b_1_04·b_1_2 + b_4_12·b_1_05 + b_4_12·b_4_13·b_1_2 + b_4_122·b_1_0 + b_4_122·a_1_1 + c_4_15·b_1_0·b_1_3·b_3_10 + c_4_15·b_1_02·b_3_10 + c_4_14·b_1_0·b_1_3·b_3_10 + c_4_14·b_1_0·b_1_34 + c_4_14·b_1_02·b_3_10 + c_4_14·b_1_02·b_1_33 + c_4_14·b_1_03·b_1_32 + c_4_14·b_1_05 + b_4_13·c_4_15·b_1_2 + b_4_13·c_4_14·b_1_2 + b_4_13·c_4_14·b_1_0 + b_4_12·c_4_14·b_1_2 + b_4_12·c_4_14·b_1_0 + b_4_12·c_4_15·a_1_1
- b_4_13·b_6_37 + b_4_12·b_1_36 + b_4_12·b_1_0·b_1_35 + b_4_12·b_1_03·b_1_33
+ b_4_12·b_1_06 + b_4_12·b_4_13·b_1_02 + b_4_122·b_1_32 + b_4_122·b_1_0·b_1_2 + b_4_122·b_1_02 + b_4_122·a_1_1·b_1_3 + c_4_14·b_1_36 + c_4_14·b_1_0·b_1_35 + c_4_14·b_1_03·b_1_33 + c_4_14·b_1_06 + b_4_13·c_4_15·b_1_2·b_1_3 + b_4_13·c_4_15·b_1_0·b_1_3 + b_4_13·c_4_15·b_1_0·b_1_2 + b_4_13·c_4_15·b_1_02 + b_4_13·c_4_14·b_1_32 + b_4_13·c_4_14·b_1_2·b_1_3 + b_4_13·c_4_14·b_1_0·b_1_3 + b_4_13·c_4_14·b_1_0·b_1_2 + b_4_12·c_4_14·b_1_32 + b_4_12·c_4_14·b_1_2·b_1_3 + b_4_12·c_4_14·b_1_0·b_1_2 + b_4_12·c_4_14·b_1_02 + b_4_12·c_4_14·a_1_1·b_1_3
- b_4_12·b_1_03·b_3_10 + b_4_12·b_1_03·b_1_33 + b_4_12·b_1_04·b_1_32
+ b_4_12·b_1_06 + b_4_12·b_6_37 + b_4_12·b_4_13·b_1_2·b_1_3 + b_4_12·b_4_13·b_1_0·b_1_3 + b_4_12·b_4_13·b_1_0·b_1_2 + b_4_122·b_1_0·b_1_2 + b_4_122·b_1_02 + c_4_15·b_1_03·b_3_10 + c_4_14·b_1_03·b_1_33 + c_4_14·b_1_04·b_1_32 + c_4_14·b_1_06 + b_4_13·c_4_15·b_1_2·b_1_3 + b_4_13·c_4_15·b_1_0·b_1_2 + b_4_13·c_4_14·b_1_32 + b_4_13·c_4_14·b_1_2·b_1_3 + b_4_13·c_4_14·b_1_02 + b_4_12·c_4_15·b_1_2·b_1_3 + b_4_12·c_4_15·b_1_0·b_1_3 + b_4_12·c_4_15·b_1_0·b_1_2 + b_4_12·c_4_15·b_1_02 + b_4_12·c_4_14·b_1_32 + b_4_12·c_4_14·b_1_2·b_1_3 + b_4_12·c_4_14·b_1_0·b_1_3 + b_4_12·c_4_15·a_1_1·b_1_3 + b_4_12·c_4_14·a_1_1·b_1_3
- b_6_372 + b_4_12·b_1_38 + b_4_12·b_1_02·b_1_36 + b_4_12·b_1_06·b_1_32
+ b_4_12·b_1_07·b_1_3 + b_4_12·b_6_37·b_1_02 + b_4_12·b_4_13·b_1_0·b_1_33 + b_4_12·b_4_13·b_1_03·b_1_3 + b_4_12·b_4_13·b_1_04 + b_4_12·b_4_132 + b_4_122·b_1_3·b_3_10 + b_4_122·b_1_34 + b_4_122·b_1_02·b_1_2·b_1_3 + b_4_122·b_1_03·b_1_2 + b_4_122·b_1_04 + b_4_123 + b_4_122·a_1_1·b_1_33 + c_4_15·b_6_37·b_1_02 + c_4_14·b_1_38 + c_4_14·b_1_02·b_1_36 + c_4_14·b_1_06·b_1_32 + c_4_14·b_1_07·b_1_3 + b_4_13·c_4_15·b_1_02·b_1_32 + b_4_13·c_4_15·b_1_03·b_1_3 + b_4_13·c_4_14·b_1_0·b_1_33 + b_4_13·c_4_14·b_1_02·b_1_32 + b_4_13·c_4_14·b_1_03·b_1_3 + b_4_13·c_4_14·b_1_04 + b_4_132·c_4_14 + b_4_12·c_4_15·b_1_0·b_3_10 + b_4_12·c_4_15·b_1_03·b_1_3 + b_4_12·c_4_14·b_1_2·b_3_10 + b_4_12·c_4_14·b_1_02·b_1_32 + b_4_12·c_4_14·b_1_02·b_1_2·b_1_3 + b_4_12·c_4_14·b_1_03·b_1_3 + b_4_122·c_4_14 + b_4_12·c_4_14·a_1_1·b_1_33 + c_4_152·b_1_02·b_1_32 + c_4_152·b_1_02·b_1_2·b_1_3 + c_4_152·b_1_03·b_1_3 + c_4_14·c_4_15·b_1_02·b_1_32 + c_4_14·c_4_15·b_1_02·b_1_2·b_1_3 + c_4_14·c_4_15·b_1_03·b_1_3 + c_4_142·b_1_02·b_1_32 + c_4_142·b_1_02·b_1_2·b_1_3 + c_4_142·b_1_03·b_1_2 + c_4_142·b_1_04
Data used for Benson′s test
- Benson′s completion test succeeded in degree 12.
- 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_14, a Duflot regular element of degree 4
- c_4_15, a Duflot regular element of degree 4
- b_1_32 + b_1_0·b_1_3 + b_1_02, an element of degree 2
- b_1_32, 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
- b_1_0 → 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_3_10 → 0, an element of degree 3
- b_4_12 → 0, an element of degree 4
- b_4_13 → 0, an element of degree 4
- c_4_14 → c_1_14, an element of degree 4
- c_4_15 → c_1_14 + c_1_04, an element of degree 4
- b_6_37 → 0, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_1 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_3_10 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_4_12 → c_1_24 + c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- b_4_13 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- c_4_14 → c_1_24 + c_1_1·c_1_23 + c_1_14, an element of degree 4
- c_4_15 → c_1_1·c_1_23 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_6_37 → c_1_1·c_1_25 + c_1_14·c_1_22, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_1 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- b_3_10 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- b_4_12 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- b_4_13 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_23 + c_1_12·c_1_32 + c_1_12·c_1_22, an element of degree 4
- c_4_14 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_23
+ c_1_02·c_1_22, an element of degree 4
- c_4_15 → c_1_1·c_1_22·c_1_3 + c_1_1·c_1_23 + c_1_12·c_1_32 + c_1_14
+ c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
- b_6_37 → c_1_1·c_1_2·c_1_34 + c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3
+ c_1_12·c_1_2·c_1_33 + c_1_12·c_1_24 + c_1_13·c_1_2·c_1_32 + c_1_13·c_1_22·c_1_3 + c_1_14·c_1_22 + c_1_0·c_1_22·c_1_33 + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_24·c_1_3 + c_1_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_2·c_1_33 + c_1_02·c_1_22·c_1_32 + 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_2·c_1_3 + c_1_04·c_1_22, an element of degree 6
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