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Mod-5-Cohomology of McL, a group of order 898128000

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• McL is a group of order 898128000.
• The group order factors as 2⁷ · 3⁶ · 5³ · 7 · 11.
• The group is defined by Group([(1,191)(2,182)(4,81)(5,55)(6,60)(8,66)(9,272)(10,177)(11,192)(12,163)(14,242)(15,133)(16,107)(18,267)(19,108)(20,218)(21,198)(22,185)(23,211)(24,82)(25,204)(26,195)(27,132)(28,253)(29,207)(30,59)(31,179)(32,154)(33,264)(34,152)(35,92)(36,189)(37,217)(38,197)(39,85)(40,156)(42,184)(43,102)(44,50)(45,216)(46,99)(47,181)(49,199)(51,111)(53,158)(54,236)(56,210)(58,103)(61,263)(63,119)(64,138)(65,127)(67,105)(68,137)(70,125)(71,144)(72,219)(73,261)(75,175)(77,269)(78,237)(79,268)(83,232)(84,256)(86,104)(87,95)(89,234)(90,233)(91,140)(94,149)(97,173)(98,160)(100,112)(101,123)(106,221)(109,131)(110,176)(113,262)(114,257)(115,201)(117,260)(118,238)(120,275)(121,214)(122,225)(124,246)(126,170)(129,141)(134,196)(135,167)(142,235)(143,224)(145,205)(146,249)(148,226)(150,243)(153,193)(155,228)(161,231)(164,215)(165,180)(168,222)(169,270)(171,241)(172,259)(174,212)(178,188)(183,266)(186,203)(190,250)(202,247)(208,255)(209,251)(213,252)(223,271)(227,274)(230,240)(239,254)(245,258)(265,273),(1,24,204,92,155)(2,28,272,165,78)(3,67,142,31,255)(4,168,77,17,100)(5,118,19,223,211)(6,274,137,79,245)(7,98,75,73,14)(8,209,43,139,193)(9,266,104,70,145)(10,271,12,13,71)(11,247,138,121,269)(15,218,133,164,196)(16,170,182,65,171)(18,26,198,124,185)(20,128,159,83,38)(21,264,27,64,162)(22,116,53,101,51)(23,179,244,80,203)(25,40,177,85,191)(29,56,135,68,195)(30,132,42,248,146)(32,148,114,49,134)(33,61,163,90,227)(34,241,233,95,181)(35,89,82,205,41)(36,239,275,257,183)(37,54,99,249,176)(39,126,189,136,230)(44,172,153,125,119)(45,234,222,232,212)(46,214,69,167,190)(47,140,268,174,62)(48,55,113,220,235)(50,213,224,202,130)(52,107,262,226,88)(57,221,261,129,58)(59,252,260,216,166)(60,84,188,208,201)(63,94,173,210,81)(66,192,93,169,110)(72,152,197,217,254)(74,206,154,186,219)(76,91,180,238,112)(86,215,231,131,225)(87,115,158,178,240)(96,265,161,120,144)(97,250,243,263,109)(102,207,246,122,127)(103,156,160,151,150)(105,267,199,111,117)(106,273,242,149,143)(108,256,157,147,184)(123,141,259,175,200)(187,251,258,194,236)(228,237,253,270,229)]).
• It is non-abelian.
• It has 5-Rank 2.
• The centre of a Sylow 5-subgroup has rank 1.
• Its Sylow 5-subgroup 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

  1  −  2·t  +  2·t2  −  2·t3  +  3·t4  −  3·t5  +  2·t6  −  t7  +  t8  −  2·t9  +  2·t10  −  2·t11  +  3·t12  −  2·t13  +  t14  −  2·t17  +  3·t18  −  3·t19  +  3·t20  −  2·t21  +  t22  −  2·t25  +  3·t26  −  2·t27  +  2·t28  −  2·t29  +  t30  −  t31  +  2·t32  −  3·t33  +  3·t34  −  2·t35  +  2·t36  −  2·t37  +  t38

  ( − 1  +  t)2 · (1  +  t2) · (1  −  t  +  t2  −  t3  +  t4) · (1  +  t4) · (1  +  t  +  t2  +  t3  +  t4) · (1  −  t2  +  t4  −  t6  +  t8) · (1  −  t4  +  t8  −  t12  +  t16)

• The a-invariants are -∞,-16,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
• The filter degree type of any filter regular HSOP is [-1, -2, -2].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 12 minimal generators of maximal degree 40:

1. a_4_0, a nilpotent element of degree 4
2. a_5_0, a nilpotent element of degree 5
3. a_7_0, a nilpotent element of degree 7
4. b_8_0, an element of degree 8
5. a_13_1, a nilpotent element of degree 13
6. b_14_0, an element of degree 14
7. a_15_1, a nilpotent element of degree 15
8. a_16_1, a nilpotent element of degree 16
9. a_23_1, a nilpotent element of degree 23
10. a_24_1, a nilpotent element of degree 24
11. a_39_1, a nilpotent element of degree 39
12. c_40_1, a Duflot element of degree 40

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 6 "obvious" relations:
   a_5_0², a_7_0², a_13_1², a_15_1², a_23_1², a_39_1²

Apart from that, there are 52 minimal relations of maximal degree 63:

1. a_4_0²
2. a_4_0·a_5_0
3. a_4_0·a_7_0
4. a_4_0·b_8_0 + 2·a_5_0·a_7_0
5. a_4_0·a_13_1
6. a_4_0·b_14_0 + a_5_0·a_13_1
7. a_4_0·a_15_1
8. a_4_0·a_16_1
9. a_5_0·a_15_1
10. a_7_0·a_13_1
11. a_16_1·a_5_0
12. b_14_0·a_7_0 + 2·b_8_0·a_13_1
13. a_7_0·a_15_1
14. a_16_1·a_7_0
15. b_8_0·a_15_1
16. b_8_0·a_16_1
17. a_4_0·a_23_1
18. a_4_0·a_24_1
19. a_5_0·a_23_1
20. a_13_1·a_15_1
21. a_16_1·a_13_1
22. a_24_1·a_5_0
23. b_14_0·a_15_1
24. a_7_0·a_23_1
25. b_14_0·a_16_1
26. a_16_1·a_15_1
27. a_24_1·a_7_0
28. b_8_0·a_23_1
29. a_16_1²
30. b_8_0·a_24_1
31. a_13_1·a_23_1
32. a_24_1·a_13_1
33. b_14_0·a_23_1
34. a_15_1·a_23_1
35. b_14_0·a_24_1
36. a_16_1·a_23_1
37. a_24_1·a_15_1
38. a_16_1·a_24_1
39. a_4_0·a_39_1
40. a_5_0·a_39_1
41. a_7_0·a_39_1 + 2·b_14_0²·a_5_0·a_13_1
42. a_24_1·a_23_1
43. b_14_0³·a_5_0 + b_8_0·a_39_1
44. a_24_1²
45. a_13_1·a_39_1 + c_40_1·a_5_0·a_7_0
46. b_14_0·a_39_1 + 2·b_8_0·c_40_1·a_5_0
47. a_15_1·a_39_1
48. a_16_1·a_39_1
49. b_14_0³·a_13_1 + b_8_0·c_40_1·a_7_0
50. b_14_0⁴ + b_8_0³·b_14_0·a_5_0·a_13_1 − 2·b_8_0²·c_40_1
51. a_23_1·a_39_1
52. a_24_1·a_39_1

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 63 using the Hilbert-Poincaré criterion.
• 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_40_1, an element of degree 40
  2. b_8_0, an element of degree 8
• A Duflot regular sequence is given by c_40_1.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, 24, 46].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(Normalizer(McL,Centre(SylowSubgroup(McL,5))); GF(5))

1. a_4_0 → a_4_0
2. a_5_0 → a_5_0
3. a_7_0 → a_7_0
4. b_8_0 → b_8_0
5. a_13_1 → a_13_1
6. b_14_0 → b_14_0
7. a_15_1 → a_15_1
8. a_16_1 → a_16_1
9. a_23_1 → a_23_1
10. a_24_1 → a_24_1
11. a_39_1 → a_39_1
12. c_40_1 → c_40_1

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1

1. a_4_0 → 0, an element of degree 4
2. a_5_0 → 0, an element of degree 5
3. a_7_0 → 0, an element of degree 7
4. b_8_0 → 0, an element of degree 8
5. a_13_1 → 0, an element of degree 13
6. b_14_0 → 0, an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 → 0, an element of degree 39
12. c_40_1 →  − 2·c_2_0²⁰, an element of degree 40

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

1. a_4_0 → 2·c_2_2·a_1_0·a_1_1, an element of degree 4
2. a_5_0 →  − 2·c_2_2²·a_1_0 + 2·c_2_1·c_2_2·a_1_1, an element of degree 5
3. a_7_0 → 2·c_2_2³·a_1_1, an element of degree 7
4. b_8_0 →  − c_2_2⁴, an element of degree 8
5. a_13_1 → 2·c_2_1·c_2_2⁵·a_1_1 − 2·c_2_1⁵·c_2_2·a_1_1, an element of degree 13
6. b_14_0 → 2·c_2_1·c_2_2⁶ − 2·c_2_1⁵·c_2_2², an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 →  − c_2_1³·c_2_2¹⁶·a_1_0 + c_2_1⁴·c_2_2¹⁵·a_1_1 − 2·c_2_1⁷·c_2_2¹²·a_1_0
       + 2·c_2_1⁸·c_2_2¹¹·a_1_1 + 2·c_2_1¹¹·c_2_2⁸·a_1_0 − 2·c_2_1¹²·c_2_2⁷·a_1_1
       + c_2_1¹⁵·c_2_2⁴·a_1_0 − c_2_1¹⁶·c_2_2³·a_1_1, an element of degree 39
12. c_40_1 →  − c_2_1²·c_2_2¹⁷·a_1_0·a_1_1 + 2·c_2_1⁶·c_2_2¹³·a_1_0·a_1_1
       − c_2_1¹⁰·c_2_2⁹·a_1_0·a_1_1 − 2·c_2_1⁴·c_2_2¹⁶ − 2·c_2_1⁸·c_2_2¹²
       − 2·c_2_1¹²·c_2_2⁸ − 2·c_2_1¹⁶·c_2_2⁴ − 2·c_2_1²⁰, an element of degree 40

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

1. a_4_0 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
2. a_5_0 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
3. a_7_0 → 2·c_2_2³·a_1_1, an element of degree 7
4. b_8_0 →  − c_2_2⁴, an element of degree 8
5. a_13_1 →  − c_2_1·c_2_2⁵·a_1_1 + c_2_1⁵·c_2_2·a_1_1, an element of degree 13
6. b_14_0 →  − c_2_1·c_2_2⁶ + c_2_1⁵·c_2_2², an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 →  − c_2_1³·c_2_2¹⁶·a_1_0 + c_2_1⁴·c_2_2¹⁵·a_1_1 − 2·c_2_1⁷·c_2_2¹²·a_1_0
       + 2·c_2_1⁸·c_2_2¹¹·a_1_1 + 2·c_2_1¹¹·c_2_2⁸·a_1_0 − 2·c_2_1¹²·c_2_2⁷·a_1_1
       + c_2_1¹⁵·c_2_2⁴·a_1_0 − c_2_1¹⁶·c_2_2³·a_1_1, an element of degree 39
12. c_40_1 → 2·c_2_1²·c_2_2¹⁷·a_1_0·a_1_1 + c_2_1⁶·c_2_2¹³·a_1_0·a_1_1
       + 2·c_2_1¹⁰·c_2_2⁹·a_1_0·a_1_1 − 2·c_2_1⁴·c_2_2¹⁶ − 2·c_2_1⁸·c_2_2¹²
       − 2·c_2_1¹²·c_2_2⁸ − 2·c_2_1¹⁶·c_2_2⁴ − 2·c_2_1²⁰, an element of degree 40

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

1. a_4_0 →  − c_2_2·a_1_0·a_1_1, an element of degree 4
2. a_5_0 → c_2_2²·a_1_0 − c_2_1·c_2_2·a_1_1, an element of degree 5
3. a_7_0 → 2·c_2_2³·a_1_1, an element of degree 7
4. b_8_0 →  − c_2_2⁴, an element of degree 8
5. a_13_1 →  − c_2_1·c_2_2⁵·a_1_1 + c_2_1⁵·c_2_2·a_1_1, an element of degree 13
6. b_14_0 →  − c_2_1·c_2_2⁶ + c_2_1⁵·c_2_2², an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 →  − c_2_1³·c_2_2¹⁶·a_1_0 + c_2_1⁴·c_2_2¹⁵·a_1_1 − 2·c_2_1⁷·c_2_2¹²·a_1_0
       + 2·c_2_1⁸·c_2_2¹¹·a_1_1 + 2·c_2_1¹¹·c_2_2⁸·a_1_0 − 2·c_2_1¹²·c_2_2⁷·a_1_1
       + c_2_1¹⁵·c_2_2⁴·a_1_0 − c_2_1¹⁶·c_2_2³·a_1_1, an element of degree 39
12. c_40_1 → 2·c_2_1²·c_2_2¹⁷·a_1_0·a_1_1 + c_2_1⁶·c_2_2¹³·a_1_0·a_1_1
       + 2·c_2_1¹⁰·c_2_2⁹·a_1_0·a_1_1 − 2·c_2_1⁴·c_2_2¹⁶ − 2·c_2_1⁸·c_2_2¹²
       − 2·c_2_1¹²·c_2_2⁸ − 2·c_2_1¹⁶·c_2_2⁴ − 2·c_2_1²⁰, an element of degree 40

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

1. a_4_0 →  − 2·c_2_2·a_1_0·a_1_1, an element of degree 4
2. a_5_0 → 2·c_2_2²·a_1_0 − 2·c_2_1·c_2_2·a_1_1, an element of degree 5
3. a_7_0 → 2·c_2_2³·a_1_1, an element of degree 7
4. b_8_0 →  − c_2_2⁴, an element of degree 8
5. a_13_1 →  − 2·c_2_1·c_2_2⁵·a_1_1 + 2·c_2_1⁵·c_2_2·a_1_1, an element of degree 13
6. b_14_0 →  − 2·c_2_1·c_2_2⁶ + 2·c_2_1⁵·c_2_2², an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 →  − c_2_1³·c_2_2¹⁶·a_1_0 + c_2_1⁴·c_2_2¹⁵·a_1_1 − 2·c_2_1⁷·c_2_2¹²·a_1_0
       + 2·c_2_1⁸·c_2_2¹¹·a_1_1 + 2·c_2_1¹¹·c_2_2⁸·a_1_0 − 2·c_2_1¹²·c_2_2⁷·a_1_1
       + c_2_1¹⁵·c_2_2⁴·a_1_0 − c_2_1¹⁶·c_2_2³·a_1_1, an element of degree 39
12. c_40_1 → c_2_1²·c_2_2¹⁷·a_1_0·a_1_1 − 2·c_2_1⁶·c_2_2¹³·a_1_0·a_1_1
       + c_2_1¹⁰·c_2_2⁹·a_1_0·a_1_1 − 2·c_2_1⁴·c_2_2¹⁶ − 2·c_2_1⁸·c_2_2¹²
       − 2·c_2_1¹²·c_2_2⁸ − 2·c_2_1¹⁶·c_2_2⁴ − 2·c_2_1²⁰, an element of degree 40

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

1. a_4_0 →  − 2·c_2_2·a_1_0·a_1_1, an element of degree 4
2. a_5_0 → 2·c_2_2²·a_1_0 − 2·c_2_1·c_2_2·a_1_1, an element of degree 5
3. a_7_0 → 2·c_2_2³·a_1_1, an element of degree 7
4. b_8_0 →  − c_2_2⁴, an element of degree 8
5. a_13_1 →  − 2·c_2_1·c_2_2⁵·a_1_1 + 2·c_2_1⁵·c_2_2·a_1_1, an element of degree 13
6. b_14_0 →  − 2·c_2_1·c_2_2⁶ + 2·c_2_1⁵·c_2_2², an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 →  − c_2_1³·c_2_2¹⁶·a_1_0 + c_2_1⁴·c_2_2¹⁵·a_1_1 − 2·c_2_1⁷·c_2_2¹²·a_1_0
       + 2·c_2_1⁸·c_2_2¹¹·a_1_1 + 2·c_2_1¹¹·c_2_2⁸·a_1_0 − 2·c_2_1¹²·c_2_2⁷·a_1_1
       + c_2_1¹⁵·c_2_2⁴·a_1_0 − c_2_1¹⁶·c_2_2³·a_1_1, an element of degree 39
12. c_40_1 → c_2_1²·c_2_2¹⁷·a_1_0·a_1_1 − 2·c_2_1⁶·c_2_2¹³·a_1_0·a_1_1
       + c_2_1¹⁰·c_2_2⁹·a_1_0·a_1_1 − 2·c_2_1⁴·c_2_2¹⁶ − 2·c_2_1⁸·c_2_2¹²
       − 2·c_2_1¹²·c_2_2⁸ − 2·c_2_1¹⁶·c_2_2⁴ − 2·c_2_1²⁰, an element of degree 40

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

1. a_4_0 → c_2_2·a_1_0·a_1_1, an element of degree 4
2. a_5_0 →  − c_2_2²·a_1_0 + c_2_1·c_2_2·a_1_1, an element of degree 5
3. a_7_0 → 2·c_2_2³·a_1_1, an element of degree 7
4. b_8_0 →  − c_2_2⁴, an element of degree 8
5. a_13_1 → c_2_1·c_2_2⁵·a_1_1 − c_2_1⁵·c_2_2·a_1_1, an element of degree 13
6. b_14_0 → c_2_1·c_2_2⁶ − c_2_1⁵·c_2_2², an element of degree 14
7. a_15_1 → 0, an element of degree 15
8. a_16_1 → 0, an element of degree 16
9. a_23_1 → 0, an element of degree 23
10. a_24_1 → 0, an element of degree 24
11. a_39_1 →  − c_2_1³·c_2_2¹⁶·a_1_0 + c_2_1⁴·c_2_2¹⁵·a_1_1 − 2·c_2_1⁷·c_2_2¹²·a_1_0
       + 2·c_2_1⁸·c_2_2¹¹·a_1_1 + 2·c_2_1¹¹·c_2_2⁸·a_1_0 − 2·c_2_1¹²·c_2_2⁷·a_1_1
       + c_2_1¹⁵·c_2_2⁴·a_1_0 − c_2_1¹⁶·c_2_2³·a_1_1, an element of degree 39
12. c_40_1 →  − 2·c_2_1²·c_2_2¹⁷·a_1_0·a_1_1 − c_2_1⁶·c_2_2¹³·a_1_0·a_1_1
       − 2·c_2_1¹⁰·c_2_2⁹·a_1_0·a_1_1 − 2·c_2_1⁴·c_2_2¹⁶ − 2·c_2_1⁸·c_2_2¹²
       − 2·c_2_1¹²·c_2_2⁸ − 2·c_2_1¹⁶·c_2_2⁴ − 2·c_2_1²⁰, an element of degree 40

About the group

Ring generators

Ring relations

Completion information

Restriction maps