David J. Green

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Cohomology of group number 14 of order 625

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General information on the group

• The group is also known as E125*C25, the Central product E125 * C_25.
• The group has 3 minimal generators and exponent 25.
• It is non-abelian.
• It has p-Rank 2.
• Its center has rank 1.
• It has 6 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

  (t2  +  t  +  1) · (t6  −  t5  +  t4  +  1)

  (t  −  1)2 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)

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Ring generators

The cohomology ring has 9 minimal generators of maximal degree 10:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_2, a nilpotent element of degree 1
4. b_2_3, an element of degree 2
5. b_2_4, an element of degree 2
6. a_4_3, a nilpotent element of degree 4
7. a_6_4, a nilpotent element of degree 6
8. b_8_9, an element of degree 8
9. c_10_12, a Duflot regular element of degree 10

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Ring relations

There are 3 "obvious" relations:
   a_1_0², a_1_1², a_1_2²

Apart from that, there are 19 minimal relations of maximal degree 16:

1. b_2_4·a_1_0 − b_2_3·a_1_1
2. a_4_3·a_1_1
3. a_4_3·a_1_0
4. b_2_3·a_4_3
5. b_2_4·a_4_3
6. a_6_4·a_1_1
7. a_6_4·a_1_0
8. a_4_3²
9. b_2_3·a_6_4
10. b_2_4·a_6_4
11. b_8_9·a_1_1 − 2·b_2_3·b_2_4³·a_1_1 − b_2_3²·b_2_4²·a_1_1 + b_2_3³·b_2_4·a_1_1
       − b_2_3⁴·a_1_1
12. b_8_9·a_1_0 − b_2_3·b_2_4³·a_1_1 − 2·b_2_3²·b_2_4²·a_1_1 − b_2_3³·b_2_4·a_1_1
       + b_2_3⁴·a_1_1
13. a_4_3·a_6_4
14. b_2_3·b_8_9 − b_2_3·b_2_4⁴ − 2·b_2_3²·b_2_4³ − b_2_3³·b_2_4² + b_2_3⁴·b_2_4
15. b_2_4·b_8_9 − 2·b_2_3·b_2_4⁴ − b_2_3²·b_2_4³ + b_2_3³·b_2_4² − b_2_3⁴·b_2_4
16. a_6_4²
17. a_4_3·b_8_9
18. a_6_4·b_8_9
19. b_8_9² + 2·b_2_3⁴·b_2_4⁴ − 2·b_2_3⁵·b_2_4³ − 2·b_2_3⁶·b_2_4²
       − 2·b_2_3⁷·b_2_4

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 16.
• 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:
  1. c_10_12, a Duflot regular element of degree 10
  2. b_8_9 − b_2_4⁴ − 2·b_2_3·b_2_4³ − b_2_3²·b_2_4² + b_2_3³·b_2_4 − b_2_3⁴, 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].
• We found that there exists some filter regular HSOP formed by the first term of the above HSOP, together with 1 elements of degree 2.

About the group

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 → 0, an element of degree 8
9. c_10_12 → c_2_0⁵, an element of degree 10

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → a_1_1, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 → c_2_2, an element of degree 2
5. b_2_4 → 0, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 → 0, an element of degree 8
9. c_10_12 →  − c_2_1·c_2_2⁴ + c_2_1⁵, an element of degree 10

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. b_2_4 → c_2_2, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 → 0, an element of degree 8
9. c_10_12 →  − c_2_1·c_2_2⁴ + c_2_1⁵, an element of degree 10

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → a_1_1, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 → c_2_2, an element of degree 2
5. b_2_4 → c_2_2, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 →  − 2·c_2_2⁴, an element of degree 8
9. c_10_12 →  − c_2_1·c_2_2⁴ + c_2_1⁵, an element of degree 10

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → 2·a_1_1, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 → 2·c_2_2, an element of degree 2
5. b_2_4 → c_2_2, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 → c_2_2⁴, an element of degree 8
9. c_10_12 →  − c_2_1·c_2_2⁴ + c_2_1⁵, an element of degree 10

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 → a_1_1, an element of degree 1
2. a_1_1 → 2·a_1_1, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 → c_2_2, an element of degree 2
5. b_2_4 → 2·c_2_2, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 →  − c_2_2⁴, an element of degree 8
9. c_10_12 →  − c_2_2⁵ − c_2_1·c_2_2⁴ + c_2_1⁵, an element of degree 10

Restriction map to a maximal el. ab. subgp. of rank 2

1. a_1_0 →  − a_1_1, an element of degree 1
2. a_1_1 → a_1_1, an element of degree 1
3. a_1_2 → 0, an element of degree 1
4. b_2_3 →  − c_2_2, an element of degree 2
5. b_2_4 → c_2_2, an element of degree 2
6. a_4_3 → 0, an element of degree 4
7. a_6_4 → 0, an element of degree 6
8. b_8_9 → c_2_2⁴, an element of degree 8
9. c_10_12 →  − 2·c_2_2⁵ − c_2_1·c_2_2⁴ + c_2_1⁵, an element of degree 10

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 625