## Cohomology of group number 3 of order 125

### General information on the group

• The group is also known as E125, the Extraspecial 5-group of order 125 and exponent 5.
• The group has 2 minimal generators and exponent 5.
• 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  +  1) · (t6  +  t2  +  1) (t  −  1)2 · (t4  −  t3  +  t2  −  t  +  1) · (t4  +  t3  +  t2  +  t  +  1)
• The a-invariants are -∞,-4,-2. They were obtained using the filter regular HSOP of the Benson test.

#### Ring generators

The cohomology ring has 12 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_2_0, a nilpotent element of degree 2
4. a_2_1, a nilpotent element of degree 2
5. b_2_2, an element of degree 2
6. b_2_3, an element of degree 2
7. a_3_4, a nilpotent element of degree 3
8. a_3_5, a nilpotent element of degree 3
9. a_7_8, a nilpotent element of degree 7
10. b_8_9, an element of degree 8
11. a_9_11, a nilpotent element of degree 9
12. c_10_12, a Duflot regular element of degree 10

#### Ring relations

There are 6 "obvious" relations:
a_1_02, a_1_12, a_3_42, a_3_52, a_7_82, a_9_112

Apart from that, there are 44 minimal relations of maximal degree 17:

1. a_1_0·a_1_1
2. a_2_0·a_1_0
3. a_2_1·a_1_1
4. a_2_1·a_1_0 − a_2_0·a_1_1
5. b_2_3·a_1_0 − b_2_2·a_1_1
6. a_2_02
7. a_2_0·a_2_1
8. a_2_12
9.  − 2·a_2_1·b_2_2 − a_2_0·b_2_3 + a_1_1·a_3_4
10. a_2_0·b_2_2 + a_1_0·a_3_4
11.  − a_2_1·b_2_3 + a_1_1·a_3_5
12. a_2_1·b_2_2 + 2·a_2_0·b_2_3 + a_1_0·a_3_5
13. a_2_0·a_3_4
14.  − b_2_3·a_3_4 + b_2_2·a_3_5 + 2·b_2_2·b_2_3·a_1_1 + 2·b_2_22·a_1_1
15.  − a_2_1·a_3_4 + a_2_0·a_3_5
16. a_2_1·a_3_5
17. a_1_1·a_7_8 + 2·b_2_2·b_2_3·a_1_1·a_3_5 − 2·b_2_22·a_1_1·a_3_5
+ 2·b_2_22·a_1_0·a_3_5
18. a_1_0·a_7_8 + 2·b_2_22·a_1_1·a_3_5 − 2·b_2_22·a_1_0·a_3_5 + 2·b_2_22·a_1_0·a_3_4
19. b_2_2·a_7_8 + 2·b_2_2·b_2_33·a_1_1 + 2·b_2_22·b_2_3·a_3_5 + b_2_22·b_2_32·a_1_1
− 2·b_2_23·a_3_5 + 2·b_2_23·a_3_4 + 2·b_2_23·b_2_3·a_1_1 − b_2_24·a_1_1
20. a_2_0·a_7_8
21. a_2_1·a_7_8
22. b_2_3·a_7_8 + 2·b_2_2·b_2_32·a_3_5 + b_2_2·b_2_33·a_1_1 − 2·b_2_22·b_2_3·a_3_5
+ 2·b_2_22·b_2_32·a_1_1 + 2·b_2_23·a_3_5 − 2·b_2_23·b_2_3·a_1_1 + b_2_24·a_1_1
23. b_8_9·a_1_1 + b_2_2·b_2_33·a_1_1 − 2·b_2_22·b_2_32·a_1_1 − b_2_24·a_1_1
24. b_8_9·a_1_0 − b_2_2·b_2_33·a_1_1 + b_2_22·b_2_32·a_1_1 − 2·b_2_23·b_2_3·a_1_1
25. a_3_5·a_7_8 − b_2_2·b_2_32·a_1_1·a_3_5 − 2·b_2_22·b_2_3·a_1_1·a_3_5
+ 2·b_2_23·a_1_1·a_3_5 − b_2_23·a_1_0·a_3_5
26. a_3_4·a_7_8 − 2·b_2_2·b_2_32·a_1_1·a_3_5 − 2·b_2_22·b_2_3·a_1_1·a_3_5
− 2·b_2_23·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_5
27. b_2_2·b_8_9 − b_2_2·b_2_34 + b_2_22·b_2_33 − 2·b_2_23·b_2_32
+ b_2_22·b_2_3·a_1_1·a_3_5 + b_2_23·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_4
28. a_2_0·b_8_9 + b_2_2·b_2_32·a_1_1·a_3_5 − b_2_22·b_2_3·a_1_1·a_3_5
+ 2·b_2_23·a_1_1·a_3_5
29. a_2_1·b_8_9 + b_2_2·b_2_32·a_1_1·a_3_5 − 2·b_2_22·b_2_3·a_1_1·a_3_5
− b_2_23·a_1_0·a_3_5
30. b_2_3·b_8_9 + b_2_2·b_2_34 − 2·b_2_22·b_2_33 − b_2_24·b_2_3
+ b_2_2·b_2_32·a_1_1·a_3_5 + b_2_22·b_2_3·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_5
31. a_1_1·a_9_11 − b_2_33·a_1_1·a_3_5 − 2·b_2_2·b_2_32·a_1_1·a_3_5
− 2·b_2_23·a_1_0·a_3_5
32. a_1_0·a_9_11 − 2·b_2_22·b_2_3·a_1_1·a_3_5 + 2·b_2_23·a_1_0·a_3_4
33. b_8_9·a_3_5 + b_2_2·b_2_33·a_3_5 − 2·b_2_22·b_2_32·a_3_5 − b_2_24·a_3_5
34. b_8_9·a_3_4 − b_2_2·b_2_33·a_3_5 + b_2_22·b_2_32·a_3_5 − 2·b_2_23·b_2_3·a_3_5
− 2·b_2_23·b_2_32·a_1_1 + b_2_24·b_2_3·a_1_1 − 2·b_2_25·a_1_1
35. b_2_2·a_9_11 − 2·b_2_22·b_2_32·a_3_5 + b_2_22·b_2_33·a_1_1
− 2·b_2_23·b_2_32·a_1_1 + 2·b_2_24·a_3_4 − b_2_24·b_2_3·a_1_1 − b_2_25·a_1_1
36. a_2_0·a_9_11
37. a_2_1·a_9_11
38. b_2_3·a_9_11 − b_2_34·a_3_5 − 2·b_2_2·b_2_33·a_3_5 − 2·b_2_22·b_2_33·a_1_1
− b_2_23·b_2_32·a_1_1 − 2·b_2_24·a_3_5 − 2·b_2_24·b_2_3·a_1_1
39. a_3_5·a_9_11 + 2·b_2_22·b_2_32·a_1_1·a_3_5 + b_2_23·b_2_3·a_1_1·a_3_5
+ 2·b_2_24·a_1_1·a_3_5
40. a_3_4·a_9_11 − 2·b_2_23·b_2_3·a_1_1·a_3_5 + b_2_24·a_1_1·a_3_5 + b_2_24·a_1_0·a_3_5
41. b_8_9·a_7_8 − 2·b_2_23·b_2_33·a_3_5 − b_2_24·b_2_32·a_3_5
+ 2·b_2_24·b_2_33·a_1_1 + b_2_25·b_2_3·a_3_5 + b_2_25·b_2_32·a_1_1 + b_2_26·a_3_5
− 2·b_2_26·b_2_3·a_1_1 − 2·b_2_27·a_1_1
42. b_8_92 + 2·b_2_25·b_2_33 − b_2_27·b_2_3 − 2·b_2_24·b_2_32·a_1_1·a_3_5
− 2·b_2_25·b_2_3·a_1_1·a_3_5 − 2·b_2_26·a_1_1·a_3_5 + 2·b_2_26·a_1_0·a_3_5
43. a_7_8·a_9_11 + b_2_24·b_2_32·a_1_1·a_3_5 − b_2_25·b_2_3·a_1_1·a_3_5
44. b_8_9·a_9_11 + 2·b_2_24·b_2_33·a_3_5 + b_2_25·b_2_32·a_3_5 + b_2_26·b_2_3·a_3_5
− b_2_26·b_2_32·a_1_1 + b_2_27·a_3_5 + b_2_27·b_2_3·a_1_1

### Data used for Benson′s test

• Benson′s completion test succeeded in degree 17.
• 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_34 − b_2_2·b_2_33 + 2·b_2_22·b_2_32 + b_2_24, an element of degree 8
• The Raw Filter Degree Type of that HSOP is [-1, 6, 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

1. a_1_00, an element of degree 1
2. a_1_10, an element of degree 1
3. a_2_00, an element of degree 2
4. a_2_10, an element of degree 2
5. b_2_20, an element of degree 2
6. b_2_30, an element of degree 2
7. a_3_40, an element of degree 3
8. a_3_50, an element of degree 3
9. a_7_80, an element of degree 7
10. b_8_90, an element of degree 8
11. a_9_110, an element of degree 9
12. 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_0a_1_1, an element of degree 1
2. a_1_10, an element of degree 1
3. a_2_0 − a_1_0·a_1_1, an element of degree 2
4. a_2_10, an element of degree 2
5. b_2_2c_2_2, an element of degree 2
6. b_2_30, an element of degree 2
7. a_3_4 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. a_3_50, an element of degree 3
9. a_7_82·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
10. b_8_9 − 2·c_2_23·a_1_0·a_1_1, an element of degree 8
11. a_9_112·c_2_24·a_1_0 − 2·c_2_1·c_2_23·a_1_1, an element of degree 9
12. c_10_12 − 2·c_2_24·a_1_0·a_1_1 + 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_00, an element of degree 1
2. a_1_1a_1_1, an element of degree 1
3. a_2_00, an element of degree 2
4. a_2_1a_1_0·a_1_1, an element of degree 2
5. b_2_20, an element of degree 2
6. b_2_3c_2_2, an element of degree 2
7. a_3_40, an element of degree 3
8. a_3_5 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
9. a_7_80, an element of degree 7
10. b_8_90, an element of degree 8
11. a_9_11 − c_2_24·a_1_0 + c_2_1·c_2_23·a_1_1, an element of degree 9
12. c_10_12 − 2·c_2_24·a_1_0·a_1_1 + 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_0a_1_1, an element of degree 1
2. a_1_1a_1_1, an element of degree 1
3. a_2_0 − a_1_0·a_1_1, an element of degree 2
4. a_2_1a_1_0·a_1_1, an element of degree 2
5. b_2_2c_2_2, an element of degree 2
6. b_2_3c_2_2, an element of degree 2
7. a_3_42·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. a_3_5 − 2·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
9. a_7_82·c_2_23·a_1_1 + 2·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
10. b_8_9c_2_23·a_1_0·a_1_1 + 2·c_2_24, an element of degree 8
11. a_9_110, an element of degree 9
12. c_10_122·c_2_24·a_1_0·a_1_1 + 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_02·a_1_1, an element of degree 1
2. a_1_1a_1_1, an element of degree 1
3. a_2_0 − 2·a_1_0·a_1_1, an element of degree 2
4. a_2_1a_1_0·a_1_1, an element of degree 2
5. b_2_22·c_2_2, an element of degree 2
6. b_2_3c_2_2, an element of degree 2
7. a_3_4 − c_2_2·a_1_1 − 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1, an element of degree 3
8. a_3_5c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
9. a_7_8 − 2·c_2_23·a_1_1 + 2·c_2_23·a_1_0 − 2·c_2_1·c_2_22·a_1_1, an element of degree 7
10. b_8_92·c_2_23·a_1_0·a_1_1 + 2·c_2_24, an element of degree 8
11. a_9_11 − 2·c_2_24·a_1_0 + 2·c_2_1·c_2_23·a_1_1, an element of degree 9
12. c_10_122·c_2_24·a_1_0·a_1_1 − 2·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_0a_1_1, an element of degree 1
2. a_1_12·a_1_1, an element of degree 1
3. a_2_0 − a_1_0·a_1_1, an element of degree 2
4. a_2_12·a_1_0·a_1_1, an element of degree 2
5. b_2_2c_2_2, an element of degree 2
6. b_2_32·c_2_2, an element of degree 2
7. a_3_4 − c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
8. a_3_5c_2_2·a_1_1 − 2·c_2_2·a_1_0 + 2·c_2_1·a_1_1, an element of degree 3
9. a_7_8 − c_2_23·a_1_1 + c_2_23·a_1_0 − c_2_1·c_2_22·a_1_1, an element of degree 7
10. b_8_9c_2_23·a_1_0·a_1_1 + c_2_24, an element of degree 8
11. a_9_11c_2_24·a_1_1 + c_2_24·a_1_0 − c_2_1·c_2_23·a_1_1, an element of degree 9
12. c_10_12c_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_1a_1_1, an element of degree 1
3. a_2_0a_1_0·a_1_1, an element of degree 2
4. a_2_1a_1_0·a_1_1, an element of degree 2
5. b_2_2 − c_2_2, an element of degree 2
6. b_2_3c_2_2, an element of degree 2
7. a_3_4 − 2·c_2_2·a_1_1 + c_2_2·a_1_0 − c_2_1·a_1_1, an element of degree 3
8. a_3_52·c_2_2·a_1_1 − c_2_2·a_1_0 + c_2_1·a_1_1, an element of degree 3
9. a_7_8 − 2·c_2_23·a_1_1 − c_2_23·a_1_0 + c_2_1·c_2_22·a_1_1, an element of degree 7
10. b_8_9 − 2·c_2_23·a_1_0·a_1_1 − c_2_24, an element of degree 8
11. a_9_11 − c_2_24·a_1_0 + c_2_1·c_2_23·a_1_1, an element of degree 9
12. c_10_12 − 2·c_2_24·a_1_0·a_1_1 − c_2_25 + c_2_1·c_2_24 − c_2_15, an element of degree 10

Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany

 E-mail: simon dot king at uni hyphen jena dot de Tel: +49 (0)3641 9-46184 Fax: +49 (0)3641 9-46162 Office: Zi. 3524, Ernst-Abbe-Platz 2
 E-mail: david dot green at uni hyphen jena dot de Tel: +49 3641 9-46166 Fax: +49 3641 9-46162 Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009