Simon King
David J. Green
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Singular
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Cohomology of group number 9 of order 625
General information on the group
- The group has 2 minimal generators and exponent 25.
- It is non-abelian.
- It has p-Rank 2.
- Its center has rank 1.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 2.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 2 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t8 − t7 + t6 − t5 + t4 − t3 + t2 + 1 |
| (t − 1)2 · (t4 − t3 + t2 − t + 1) · (t4 + t3 + t2 + t + 1) |
Ring generators
The cohomology ring has 23 minimal generators of maximal degree 15:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_2_0, a nilpotent element of degree 2
- a_2_1, a nilpotent element of degree 2
- b_2_2, an element of degree 2
- a_3_2, a nilpotent element of degree 3
- a_4_0, a nilpotent element of degree 4
- a_4_1, a nilpotent element of degree 4
- a_5_2, a nilpotent element of degree 5
- a_6_0, a nilpotent element of degree 6
- a_6_2, a nilpotent element of degree 6
- a_7_2, a nilpotent element of degree 7
- a_8_0, a nilpotent element of degree 8
- b_8_2, an element of degree 8
- a_9_2, a nilpotent element of degree 9
- a_9_3, a nilpotent element of degree 9
- b_10_3, an element of degree 10
- c_10_4, a Duflot regular element of degree 10
- a_11_5, a nilpotent element of degree 11
- b_12_6, an element of degree 12
- a_13_6, a nilpotent element of degree 13
- b_14_6, an element of degree 14
- a_15_6, a nilpotent element of degree 15
Ring relations
There are 10 "obvious" relations:
a_1_02, a_1_12, a_3_22, a_5_22, a_7_22, a_9_22, a_9_32, a_11_52, a_13_62, a_15_62
Apart from that, there are 220 minimal relations of maximal degree 29:
- a_1_0·a_1_1
- a_2_0·a_1_0
- a_2_1·a_1_1 + 2·a_2_0·a_1_1
- a_2_1·a_1_0 − a_2_0·a_1_1
- b_2_2·a_1_1
- a_2_02
- a_2_0·a_2_1
- a_2_12
- a_2_0·b_2_2
- a_2_1·b_2_2
- a_1_1·a_3_2
- a_1_0·a_3_2
- b_2_2·a_3_2
- a_2_0·a_3_2
- a_2_1·a_3_2
- a_4_0·a_1_1
- a_4_0·a_1_0
- a_4_1·a_1_0
- b_2_2·a_4_0
- a_2_0·a_4_0
- a_2_1·a_4_0
- b_2_2·a_4_1
- a_2_0·a_4_1
- a_2_1·a_4_1
- a_1_1·a_5_2
- a_1_0·a_5_2
- a_4_0·a_3_2
- a_4_1·a_3_2
- b_2_2·a_5_2
- a_2_0·a_5_2
- a_2_1·a_5_2
- a_6_0·a_1_1
- a_6_0·a_1_0
- a_6_2·a_1_0
- a_4_02
- a_4_0·a_4_1
- a_4_12
- a_3_2·a_5_2
- b_2_2·a_6_0
- a_2_0·a_6_0
- a_2_1·a_6_0
- b_2_2·a_6_2
- a_2_0·a_6_2
- a_2_1·a_6_2
- a_1_1·a_7_2
- a_1_0·a_7_2
- a_4_0·a_5_2
- a_4_1·a_5_2
- a_6_0·a_3_2
- a_6_2·a_3_2
- b_2_2·a_7_2
- a_2_0·a_7_2
- a_2_1·a_7_2
- a_8_0·a_1_1
- a_8_0·a_1_0
- b_8_2·a_1_0
- a_4_0·a_6_0
- a_4_1·a_6_0
- a_4_0·a_6_2
- a_4_1·a_6_2
- a_3_2·a_7_2
- a_2_0·a_8_0
- a_2_1·a_8_0
- b_2_2·b_8_2 − 2·b_2_2·a_8_0
- a_2_0·b_8_2
- − a_2_1·b_8_2 + a_1_1·a_9_2
- a_1_0·a_9_2
- − 2·a_2_1·b_8_2 + a_1_1·a_9_3
- − b_2_2·a_8_0 + a_1_0·a_9_3
- a_6_0·a_5_2
- a_6_2·a_5_2
- a_4_0·a_7_2
- a_4_1·a_7_2
- a_8_0·a_3_2
- b_2_2·a_9_2
- a_2_0·a_9_2
- a_2_1·a_9_2
- a_2_0·a_9_3
- a_2_1·a_9_3
- b_10_3·a_1_1 − b_8_2·a_3_2
- b_10_3·a_1_0
- a_6_02
- a_6_22
- a_6_0·a_6_2
- a_5_2·a_7_2
- a_4_0·a_8_0
- a_4_1·a_8_0
- a_4_1·b_8_2 − 2·a_4_0·b_8_2
- − a_4_0·b_8_2 + a_3_2·a_9_2
- − 2·a_4_0·b_8_2 + a_3_2·a_9_3
- b_2_2·b_10_3 + 2·b_2_2·a_1_0·a_9_3
- a_2_0·b_10_3
- − a_4_0·b_8_2 + a_2_1·b_10_3
- − 2·a_4_0·b_8_2 + a_1_1·a_11_5
- a_1_0·a_11_5 − b_2_2·a_1_0·a_9_3
- a_6_2·a_7_2
- a_6_0·a_7_2
- a_8_0·a_5_2
- a_4_0·a_9_2
- a_4_1·a_9_2
- a_4_0·a_9_3
- a_4_1·a_9_3
- b_10_3·a_3_2 − b_8_2·a_5_2
- b_2_2·a_11_5 − b_2_22·a_9_3 − 2·b_2_2·c_10_4·a_1_0
- a_2_0·a_11_5 + 2·a_2_0·c_10_4·a_1_1
- a_2_1·a_11_5 − a_2_0·c_10_4·a_1_1
- b_12_6·a_1_1 − b_8_2·a_5_2
- b_12_6·a_1_0 − 2·b_2_2·c_10_4·a_1_0 − 2·a_2_0·c_10_4·a_1_1
- a_6_2·a_8_0
- a_6_0·a_8_0
- a_6_2·b_8_2 + a_6_0·b_8_2
- − 2·a_6_2·b_8_2 + a_5_2·a_9_2
- a_6_2·b_8_2 + a_5_2·a_9_3
- − 2·a_6_2·b_8_2 + a_4_0·b_10_3
- a_6_2·b_8_2 + a_4_1·b_10_3
- a_6_2·b_8_2 + a_3_2·a_11_5
- b_2_2·b_12_6 − 2·b_2_22·a_1_0·a_9_3 − 2·b_2_22·c_10_4
- a_2_0·b_12_6
- − 2·a_6_2·b_8_2 + a_2_1·b_12_6
- 2·a_6_2·b_8_2 + a_1_1·a_13_6
- a_1_0·a_13_6 − b_2_22·a_1_0·a_9_3
- a_8_0·a_7_2
- a_6_2·a_9_2
- a_6_0·a_9_2
- a_6_2·a_9_3
- a_6_0·a_9_3
- b_10_3·a_5_2 + 2·b_8_2·a_7_2
- a_4_0·a_11_5
- a_4_1·a_11_5 + 2·a_4_1·c_10_4·a_1_1
- b_12_6·a_3_2 + 2·b_8_2·a_7_2
- b_2_2·a_13_6 − b_2_23·a_9_3
- a_2_0·a_13_6
- a_2_1·a_13_6
- b_14_6·a_1_1 + 2·b_8_2·a_7_2 + 2·a_4_1·c_10_4·a_1_1
- b_14_6·a_1_0 + 2·b_2_22·c_10_4·a_1_0
- a_8_02
- − a_8_0·b_8_2 + a_7_2·a_9_2
- − 2·a_8_0·b_8_2 + a_7_2·a_9_3
- a_8_0·b_8_2 + a_6_2·b_10_3
- − a_8_0·b_8_2 + a_6_0·b_10_3
- − a_8_0·b_8_2 + a_5_2·a_11_5
- 2·a_8_0·b_8_2 + a_4_0·b_12_6
- − a_8_0·b_8_2 + a_4_1·b_12_6
- − 2·a_8_0·b_8_2 + a_3_2·a_13_6
- b_2_2·b_14_6 − b_2_23·a_1_0·a_9_3 + 2·b_2_23·c_10_4
- a_2_0·b_14_6
- 2·a_8_0·b_8_2 + a_2_1·b_14_6
- 2·a_8_0·b_8_2 + a_1_1·a_15_6
- a_1_0·a_15_6 − b_2_23·a_1_0·a_9_3
- a_8_0·a_9_2
- a_8_0·a_9_3
- b_8_2·a_9_3 − 2·b_8_2·a_9_2 − 2·b_8_22·a_1_1
- b_10_3·a_7_2 − b_8_22·a_1_1
- a_6_2·a_11_5 + 2·a_6_2·c_10_4·a_1_1
- a_6_0·a_11_5
- b_12_6·a_5_2 + 2·b_8_22·a_1_1
- a_4_0·a_13_6
- a_4_1·a_13_6
- b_14_6·a_3_2 + 2·b_8_22·a_1_1
- b_2_2·a_15_6 − b_2_24·a_9_3 − b_2_23·c_10_4·a_1_0
- a_2_0·a_15_6
- a_2_1·a_15_6
- a_9_2·a_9_3 + 2·b_8_2·a_1_1·a_9_2
- a_8_0·b_10_3 − b_8_2·a_1_1·a_9_2
- a_7_2·a_11_5 − 2·b_8_2·a_1_1·a_9_2
- a_6_2·b_12_6 + b_8_2·a_1_1·a_9_2
- a_6_0·b_12_6 − b_8_2·a_1_1·a_9_2
- a_5_2·a_13_6 − 2·b_8_2·a_1_1·a_9_2
- a_4_0·b_14_6 + 2·b_8_2·a_1_1·a_9_2
- a_4_1·b_14_6 − b_8_2·a_1_1·a_9_2
- a_3_2·a_15_6 + 2·b_8_2·a_1_1·a_9_2
- 2·b_10_3·a_9_3 + b_10_3·a_9_2 + b_8_22·a_3_2
- a_8_0·a_11_5
- − b_10_3·a_9_3 + b_8_2·a_11_5 − b_8_22·a_3_2 + 2·b_8_2·c_10_4·a_1_1
- b_12_6·a_7_2 − b_8_22·a_3_2
- a_6_2·a_13_6
- a_6_0·a_13_6
- b_14_6·a_5_2 + 2·b_8_22·a_3_2
- a_4_0·a_15_6
- a_4_1·a_15_6
- a_9_3·a_11_5 + a_9_2·a_11_5 − c_10_4·a_1_1·a_9_2 + 2·c_10_4·a_1_0·a_9_3
- a_9_3·a_11_5 + b_8_2·a_1_1·a_11_5 + c_10_4·a_1_1·a_9_2 + 2·c_10_4·a_1_0·a_9_3
- a_8_0·b_12_6 − 2·a_9_3·a_11_5 − 2·c_10_4·a_1_1·a_9_2 − c_10_4·a_1_0·a_9_3
- − b_10_32 + b_8_2·b_12_6 + a_9_3·a_11_5 − c_10_4·a_1_1·a_9_2 − 2·c_10_4·a_1_0·a_9_3
- 2·a_9_3·a_11_5 + a_7_2·a_13_6 + 2·c_10_4·a_1_1·a_9_2 − c_10_4·a_1_0·a_9_3
- a_6_2·b_14_6 + 2·a_9_3·a_11_5 + 2·c_10_4·a_1_1·a_9_2 − c_10_4·a_1_0·a_9_3
- a_6_0·b_14_6 − 2·a_9_3·a_11_5 − 2·c_10_4·a_1_1·a_9_2 + c_10_4·a_1_0·a_9_3
- − a_9_3·a_11_5 + a_5_2·a_15_6 − c_10_4·a_1_1·a_9_2 − 2·c_10_4·a_1_0·a_9_3
- b_12_6·a_9_3 − b_10_3·a_11_5 + b_8_22·a_5_2 − 2·b_8_2·c_10_4·a_3_2
− 2·b_2_2·c_10_4·a_9_3
- b_12_6·a_9_2 + 2·b_10_3·a_11_5 − b_8_22·a_5_2 − b_8_2·c_10_4·a_3_2
- a_8_0·a_13_6
- − 2·b_10_3·a_11_5 + b_8_2·a_13_6 + b_8_22·a_5_2 − b_8_2·c_10_4·a_3_2
- b_14_6·a_7_2 − b_8_22·a_5_2
- a_6_2·a_15_6
- a_6_0·a_15_6
- a_9_2·a_13_6 + c_10_4·a_1_1·a_11_5
- 2·a_9_3·a_13_6 + b_8_2·a_1_1·a_13_6 − c_10_4·a_1_1·a_11_5
- a_8_0·b_14_6 − 2·a_9_3·a_13_6 + c_10_4·a_1_1·a_11_5 + 2·b_2_2·c_10_4·a_1_0·a_9_3
- − b_10_3·b_12_6 + b_8_2·b_14_6 − 2·c_10_4·a_1_1·a_11_5
- − 2·a_9_3·a_13_6 + a_7_2·a_15_6 + c_10_4·a_1_1·a_11_5
- − 2·b_12_6·a_11_5 + b_10_3·a_13_6 − 2·b_8_22·a_7_2 − b_8_2·c_10_4·a_5_2
− b_2_22·c_10_4·a_9_3 − 2·b_2_2·c_10_42·a_1_0 − 2·a_2_0·c_10_42·a_1_1
- b_14_6·a_9_3 − b_12_6·a_11_5 − 2·b_8_22·a_7_2 − 2·b_8_2·c_10_4·a_5_2
− b_2_22·c_10_4·a_9_3 − b_2_2·c_10_42·a_1_0 − a_2_0·c_10_42·a_1_1
- b_14_6·a_9_2 + 2·b_12_6·a_11_5 + 2·b_8_22·a_7_2 − b_8_2·c_10_4·a_5_2
+ b_2_22·c_10_4·a_9_3 + 2·b_2_2·c_10_42·a_1_0 + 2·a_2_0·c_10_42·a_1_1
- a_8_0·a_15_6
- 2·b_12_6·a_11_5 + b_8_2·a_15_6 − b_8_2·c_10_4·a_5_2 + b_2_22·c_10_4·a_9_3
+ 2·b_2_2·c_10_42·a_1_0 + 2·a_2_0·c_10_42·a_1_1
- − b_12_62 − 2·b_8_23 + c_10_4·a_1_1·a_13_6 − 2·b_2_22·c_10_4·a_1_0·a_9_3
− b_2_22·c_10_42
- − b_12_62 + b_10_3·b_14_6 + a_11_5·a_13_6 + 2·b_2_22·c_10_4·a_1_0·a_9_3
− b_2_22·c_10_42
- − b_12_62 − 2·b_8_23 − 2·a_11_5·a_13_6 + a_9_3·a_15_6 − 2·b_2_22·c_10_4·a_1_0·a_9_3
− b_2_22·c_10_42
- b_12_62 + 2·b_8_23 + 2·a_11_5·a_13_6 + a_9_2·a_15_6 − 2·b_2_22·c_10_4·a_1_0·a_9_3
+ b_2_22·c_10_42
- b_12_62 + 2·b_8_23 + 2·a_11_5·a_13_6 + b_8_2·a_1_1·a_15_6
− 2·b_2_22·c_10_4·a_1_0·a_9_3 + b_2_22·c_10_42
- b_12_6·a_13_6 − 2·b_8_22·a_9_2 − b_8_2·c_10_4·a_7_2 − 2·b_2_23·c_10_4·a_9_3
- b_14_6·a_11_5 − b_8_22·a_9_2 + b_8_23·a_1_1 + b_8_2·c_10_4·a_7_2
+ 2·b_2_23·c_10_4·a_9_3 − b_2_22·c_10_42·a_1_0 + a_4_1·c_10_42·a_1_1
- b_10_3·a_15_6 + 2·b_8_22·a_9_2 − 2·b_8_23·a_1_1
- − b_12_6·b_14_6 − 2·b_8_22·b_10_3 + a_11_5·a_15_6 + 2·b_2_23·c_10_4·a_1_0·a_9_3
+ b_2_23·c_10_42
- − 2·b_12_6·b_14_6 + b_8_22·b_10_3 + c_10_4·a_1_1·a_15_6 + b_2_23·c_10_4·a_1_0·a_9_3
+ 2·b_2_23·c_10_42
- b_14_6·a_13_6 − b_8_22·a_11_5 − 2·b_8_23·a_3_2 + 2·b_8_22·c_10_4·a_1_1
+ 2·b_2_24·c_10_4·a_9_3
- b_12_6·a_15_6 + b_8_22·a_11_5 + 2·b_8_22·c_10_4·a_1_1 − 2·b_2_24·c_10_4·a_9_3
− 2·b_2_23·c_10_42·a_1_0
- b_14_62 + 2·b_8_22·b_12_6 − 2·b_8_22·a_1_1·a_11_5 − b_2_24·c_10_4·a_1_0·a_9_3
+ b_2_24·c_10_42
- a_13_6·a_15_6 − b_8_22·a_1_1·a_11_5 − b_8_2·c_10_4·a_1_1·a_9_2
+ b_2_24·c_10_4·a_1_0·a_9_3
- b_14_6·a_15_6 − 2·b_8_22·a_13_6 − 2·b_8_23·a_5_2 − b_8_22·c_10_4·a_3_2
+ 2·b_2_25·c_10_4·a_9_3 + 2·b_2_24·c_10_42·a_1_0
Data used for Benson′s test
- Benson′s completion test succeeded in degree 29.
- 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_10_4, a Duflot regular element of degree 10
- b_8_2 + 2·b_2_24, an element of degree 8
- The Raw Filter Degree Type of that HSOP is [-1, 7, 16].
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_2_0 → 0, an element of degree 2
- a_2_1 → 0, an element of degree 2
- b_2_2 → 0, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_4_0 → 0, an element of degree 4
- a_4_1 → 0, an element of degree 4
- a_5_2 → 0, an element of degree 5
- a_6_0 → 0, an element of degree 6
- a_6_2 → 0, an element of degree 6
- a_7_2 → 0, an element of degree 7
- a_8_0 → 0, an element of degree 8
- b_8_2 → 0, an element of degree 8
- a_9_2 → 0, an element of degree 9
- a_9_3 → 0, an element of degree 9
- b_10_3 → 0, an element of degree 10
- c_10_4 → − 2·c_2_05, an element of degree 10
- a_11_5 → 0, an element of degree 11
- b_12_6 → 0, an element of degree 12
- a_13_6 → 0, an element of degree 13
- b_14_6 → 0, an element of degree 14
- a_15_6 → 0, an element of degree 15
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_0 → a_1_1, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_2_0 → 0, an element of degree 2
- a_2_1 → 0, an element of degree 2
- b_2_2 → c_2_2, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_4_0 → 0, an element of degree 4
- a_4_1 → 0, an element of degree 4
- a_5_2 → 0, an element of degree 5
- a_6_0 → 0, an element of degree 6
- a_6_2 → 0, an element of degree 6
- a_7_2 → 0, an element of degree 7
- a_8_0 → − 2·c_2_23·a_1_0·a_1_1, an element of degree 8
- b_8_2 → c_2_23·a_1_0·a_1_1, an element of degree 8
- a_9_2 → 0, an element of degree 9
- a_9_3 → 2·c_2_24·a_1_0 − 2·c_2_1·c_2_23·a_1_1, an element of degree 9
- b_10_3 → − c_2_24·a_1_0·a_1_1, an element of degree 10
- c_10_4 → c_2_24·a_1_0·a_1_1 + 2·c_2_1·c_2_24 − 2·c_2_15, an element of degree 10
- a_11_5 → 2·c_2_25·a_1_0 + 2·c_2_1·c_2_24·a_1_1 + c_2_15·a_1_1, an element of degree 11
- b_12_6 → − 2·c_2_25·a_1_0·a_1_1 − c_2_1·c_2_25 + c_2_15·c_2_2, an element of degree 12
- a_13_6 → 2·c_2_26·a_1_0 − 2·c_2_1·c_2_25·a_1_1, an element of degree 13
- b_14_6 → c_2_26·a_1_0·a_1_1 + c_2_1·c_2_26 − c_2_15·c_2_22, an element of degree 14
- a_15_6 → 2·c_2_27·a_1_0 − 2·c_2_15·c_2_22·a_1_1, an element of degree 15
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_2_0 → 0, an element of degree 2
- a_2_1 → 0, an element of degree 2
- b_2_2 → 0, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_4_0 → 0, an element of degree 4
- a_4_1 → 0, an element of degree 4
- a_5_2 → 0, an element of degree 5
- a_6_0 → 0, an element of degree 6
- a_6_2 → 0, an element of degree 6
- a_7_2 → 0, an element of degree 7
- a_8_0 → 0, an element of degree 8
- b_8_2 → 2·c_2_24, an element of degree 8
- a_9_2 → − 2·c_2_24·a_1_1, an element of degree 9
- a_9_3 → c_2_24·a_1_1, an element of degree 9
- b_10_3 → − 2·c_2_25, an element of degree 10
- c_10_4 → − c_2_25 + 2·c_2_1·c_2_24 − 2·c_2_15, an element of degree 10
- a_11_5 → − c_2_25·a_1_1, an element of degree 11
- b_12_6 → 2·c_2_26, an element of degree 12
- a_13_6 → 2·c_2_26·a_1_1, an element of degree 13
- b_14_6 → − 2·c_2_27, an element of degree 14
- a_15_6 → 2·c_2_27·a_1_1, an element of degree 15
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