Mod-2-Cohomology of group number 5603 of order 384

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • The group order factors as 27 · 3.
  • It is non-abelian.
  • It has 2-Rank 4.
  • The centre of a Sylow 2-subgroup has rank 1.
  • Its Sylow 2-subgroup has 4 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 3, 4 and 4, respectively.

Structure of the cohomology ring

The computation was based on 1 stability condition for H*(Syl2(M22); GF(2)).

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth exceeds the Duflot bound, which is 1.
  • The Poincaré series is
    1  +  2·t2  −  t3  +  t4  +  t5  +  2·t6  +  t8  +  t12
    (1  +  t) · ( − 1  +  t)4 · (1  +  t  +  t2) · (1  +  t2)2 · (1  +  t4)
  • The a-invariants are -∞,-∞,-3,-5,-4. 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, -3, -4, -4].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 14 minimal generators of maximal degree 8:

  1. b_1_1, an element of degree 1
  2. b_1_0, an element of degree 1
  3. b_2_3, an element of degree 2
  4. b_3_5, an element of degree 3
  5. b_3_0, an element of degree 3
  6. b_4_6, an element of degree 4
  7. b_4_5, an element of degree 4
  8. b_5_1, an element of degree 5
  9. b_5_0, an element of degree 5
  10. b_6_7, an element of degree 6
  11. b_7_16, an element of degree 7
  12. b_7_0, an element of degree 7
  13. b_8_7, an element of degree 8
  14. c_8_6, a Duflot element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 59 minimal relations of maximal degree 16:

  1. b_1_02·b_1_1 + b_2_3·b_1_0
  2. b_1_0·b_1_12 + b_2_3·b_1_0
  3. b_2_3·b_1_02
  4. b_1_0·b_3_5
  5. b_1_1·b_3_5
  6. b_1_02·b_3_0 + b_4_6·b_1_0
  7. b_1_0·b_1_1·b_3_0
  8. b_1_1·b_5_1 + b_1_1·b_5_0 + b_1_13·b_3_0 + b_1_0·b_5_1 + b_4_6·b_1_02
       + b_4_5·b_1_0·b_1_1
  9. b_3_0·b_3_5
  10. b_4_5·b_3_5
  11. b_4_6·b_3_5
  12. b_6_7·b_1_0 + b_4_6·b_1_03 + b_4_5·b_1_03 + b_2_3·b_5_1 + b_2_3·b_5_0
       + b_2_3·b_1_12·b_3_0
  13. b_1_02·b_5_1 + b_4_6·b_1_03 + b_2_3·b_5_1 + b_2_3·b_5_0 + b_2_3·b_1_12·b_3_0
  14. b_1_0·b_1_1·b_5_0
  15. b_1_12·b_5_0 + b_1_0·b_3_02 + b_6_7·b_1_1 + b_4_6·b_3_0 + b_4_6·b_1_13 + b_2_3·b_5_1
       + b_2_3·b_1_12·b_3_0 + b_2_3·b_4_6·b_1_1
  16. b_1_1·b_3_02 + b_1_0·b_3_02 + b_6_7·b_1_1 + b_4_6·b_3_0 + b_4_6·b_1_13 + b_2_3·b_5_1
       + b_2_3·b_4_5·b_1_1
  17. b_2_3·b_1_0·b_5_1
  18. b_2_3·b_3_02 + b_2_3·b_1_1·b_5_0 + b_2_32·b_1_1·b_3_0 + b_2_32·b_4_6 + b_2_32·b_4_5
  19. b_4_6·b_1_14 + b_4_6·b_1_0·b_3_0 + b_4_62 + b_4_5·b_1_14 + b_2_3·b_4_6·b_1_12
       + b_2_32·b_1_1·b_3_0 + b_2_32·b_4_6 + b_2_32·b_4_5
  20. b_1_0·b_7_16
  21. b_1_1·b_7_16 + b_2_3·b_4_6·b_1_12
  22. b_3_0·b_5_1 + b_1_1·b_7_0 + b_6_7·b_1_12 + b_4_6·b_1_1·b_3_0 + b_4_62
       + b_4_5·b_1_1·b_3_0 + b_4_5·b_1_14 + b_4_5·b_1_0·b_3_0 + b_4_5·b_4_6
       + b_2_3·b_1_13·b_3_0 + b_2_3·b_4_5·b_1_12
  23. b_3_5·b_5_0
  24. b_3_5·b_5_1
  25. b_6_7·b_3_5 + b_2_3·b_7_16 + b_2_32·b_4_6·b_1_1
  26. b_8_7·b_1_0 + b_4_6·b_5_1 + b_4_6·b_5_0 + b_4_6·b_1_12·b_3_0 + b_4_6·b_1_05
       + b_4_62·b_1_0 + b_4_5·b_1_05 + b_4_5·b_4_6·b_1_0
  27. b_8_7·b_1_1 + b_4_6·b_5_1 + b_4_6·b_1_12·b_3_0 + b_4_62·b_1_0 + b_2_3·b_1_14·b_3_0
       + b_2_3·b_6_7·b_1_1 + b_2_3·b_4_6·b_3_0 + b_2_3·b_4_6·b_1_13 + b_2_3·b_4_5·b_3_0
       + b_2_32·b_1_12·b_3_0
  28. b_1_0·b_3_0·b_5_0 + b_1_0·b_1_1·b_7_0 + b_4_6·b_5_1 + b_4_6·b_5_0 + b_4_6·b_1_12·b_3_0
       + b_4_62·b_1_0
  29. b_1_12·b_7_0 + b_6_7·b_3_0 + b_4_6·b_5_1 + b_4_5·b_1_12·b_3_0 + b_4_5·b_4_6·b_1_1
       + b_4_5·b_4_6·b_1_0 + b_2_3·b_7_16 + b_2_3·b_7_0 + b_2_3·b_6_7·b_1_1 + b_2_3·b_4_6·b_3_0
       + b_2_3·b_4_6·b_1_13 + b_2_3·b_4_5·b_3_0 + b_2_3·b_4_5·b_1_13
       + b_2_32·b_1_12·b_3_0 + b_2_32·b_4_6·b_1_1 + b_2_32·b_4_5·b_1_1
  30. b_4_6·b_1_1·b_5_0 + b_4_6·b_1_13·b_3_0 + b_4_6·b_6_7 + b_4_62·b_1_12
       + b_4_62·b_1_02 + b_4_5·b_1_13·b_3_0 + b_4_5·b_4_6·b_1_02 + b_2_3·b_8_7
       + b_2_3·b_4_6·b_1_1·b_3_0 + b_2_3·b_4_62 + b_2_32·b_1_1·b_5_0
       + b_2_32·b_1_13·b_3_0 + b_2_32·b_6_7 + b_2_32·b_4_6·b_1_12 + b_2_33·b_4_6
       + b_2_33·b_4_5
  31. b_3_0·b_7_16 + b_2_3·b_4_6·b_1_1·b_3_0
  32. b_3_5·b_7_16 + b_3_5·b_7_0
  33. b_5_02 + b_4_6·b_3_02 + b_4_62·b_1_02 + b_4_5·b_3_02 + b_4_5·b_4_6·b_1_02
       + b_2_3·b_6_7·b_1_12 + b_2_3·b_4_6·b_1_1·b_3_0 + b_2_3·b_4_5·b_1_1·b_3_0
       + b_2_3·b_4_5·b_1_14 + b_2_32·b_1_13·b_3_0 + c_8_6·b_1_12 + c_8_6·b_1_02
  34. b_5_0·b_5_1 + b_6_7·b_1_1·b_3_0 + b_4_6·b_1_13·b_3_0 + b_4_6·b_1_0·b_5_0
       + b_4_5·b_1_1·b_5_0 + b_4_5·b_1_0·b_5_1 + b_4_5·b_4_6·b_1_02 + b_2_3·b_1_1·b_7_0
       + b_2_3·b_6_7·b_1_12 + b_2_3·b_4_5·b_1_1·b_3_0 + b_2_3·b_4_5·b_1_14
       + b_2_3·b_4_52 + b_2_32·b_1_1·b_5_0 + b_2_32·b_1_13·b_3_0 + b_2_32·b_4_6·b_1_12
       + b_2_33·b_1_1·b_3_0 + b_2_33·b_4_6 + b_2_33·b_4_5 + c_8_6·b_1_12
       + c_8_6·b_1_0·b_1_1
  35. b_5_12 + b_6_7·b_1_14 + b_4_6·b_6_7 + b_4_5·b_1_1·b_5_0 + b_4_5·b_1_13·b_3_0
       + b_4_5·b_1_16 + b_4_5·b_1_0·b_5_1 + b_4_52·b_1_0·b_1_1 + b_2_3·b_1_15·b_3_0
       + b_2_3·b_8_7 + b_2_3·b_4_52 + b_2_32·b_1_13·b_3_0 + b_2_32·b_6_7
       + b_2_32·b_4_6·b_1_12 + b_2_32·b_4_5·b_1_12 + b_2_33·b_4_6 + b_2_33·b_4_5
       + c_8_6·b_1_12
  36. b_4_5·b_7_16 + b_2_3·b_4_5·b_4_6·b_1_1
  37. b_4_6·b_7_16 + b_2_3·b_4_62·b_1_1
  38. b_6_7·b_5_1 + b_6_7·b_5_0 + b_6_7·b_1_12·b_3_0 + b_4_6·b_1_02·b_5_0 + b_4_62·b_1_03
       + b_4_5·b_1_02·b_5_0 + b_4_5·b_4_6·b_1_03 + b_2_3·b_4_5·b_5_1 + b_2_3·b_4_5·b_5_0
       + b_2_3·b_4_5·b_1_12·b_3_0 + b_2_3·b_4_52·b_1_0 + b_2_3·c_8_6·b_1_0
  39. b_8_7·b_3_5 + b_2_32·b_7_16 + b_2_33·b_4_6·b_1_1
  40. b_1_0·b_3_0·b_7_0 + b_6_7·b_5_0 + b_4_6·b_7_0 + b_4_6·b_1_02·b_5_0
       + b_4_5·b_1_02·b_5_0 + b_4_5·b_6_7·b_1_1 + b_4_52·b_1_13 + b_2_3·b_6_7·b_1_13
       + b_2_3·b_4_6·b_5_0 + b_2_3·b_4_5·b_1_15 + b_2_3·b_4_52·b_1_1 + b_2_3·b_4_52·b_1_0
       + b_2_32·b_1_14·b_3_0 + b_2_32·b_6_7·b_1_1 + b_2_32·b_4_6·b_3_0
       + b_2_32·b_4_5·b_3_0 + b_2_32·b_4_5·b_1_13 + b_2_33·b_1_12·b_3_0 + c_8_6·b_1_13
       + b_2_3·c_8_6·b_1_1
  41. b_1_1·b_3_0·b_7_0 + b_4_6·b_1_0·b_3_02 + b_4_6·b_6_7·b_1_1 + b_4_62·b_3_0
       + b_4_62·b_1_13 + b_4_5·b_1_0·b_3_02 + b_4_5·b_4_6·b_3_0 + b_2_3·b_6_7·b_1_13
       + b_2_3·b_4_6·b_1_12·b_3_0 + b_2_3·b_4_5·b_5_1 + b_2_3·b_4_5·b_1_15
       + b_2_3·b_4_5·b_4_6·b_1_1 + b_2_3·b_4_52·b_1_0 + b_2_32·b_1_14·b_3_0
       + b_2_32·b_6_7·b_1_1 + b_2_32·b_4_6·b_1_13 + b_2_32·b_4_5·b_3_0 + b_2_33·b_5_0
       + b_2_33·b_1_12·b_3_0 + b_2_33·b_4_5·b_1_1 + c_8_6·b_1_13 + b_2_3·c_8_6·b_1_0
  42. b_3_02·b_5_0 + b_8_7·b_3_0 + b_6_7·b_5_0 + b_4_6·b_1_0·b_3_02 + b_4_6·b_1_02·b_5_0
       + b_4_6·b_6_7·b_1_1 + b_4_62·b_3_0 + b_4_62·b_1_13 + b_4_62·b_1_03
       + b_4_5·b_1_0·b_3_02 + b_4_5·b_1_02·b_5_0 + b_4_5·b_4_6·b_1_03
       + b_2_3·b_6_7·b_1_13 + b_2_3·b_4_6·b_1_12·b_3_0 + b_2_3·b_4_5·b_5_0
       + b_2_3·b_4_5·b_1_15 + b_2_32·b_7_16 + b_2_32·b_7_0 + b_2_32·b_1_14·b_3_0
       + b_2_32·b_6_7·b_1_1 + b_2_32·b_4_6·b_3_0 + b_2_32·b_4_5·b_1_13 + b_2_33·b_5_0
       + b_2_33·b_1_12·b_3_0 + b_2_33·b_4_6·b_1_1 + b_2_3·c_8_6·b_1_1 + b_2_3·c_8_6·b_1_0
  43. b_2_3·b_3_0·b_7_0 + b_2_3·b_4_6·b_1_13·b_3_0 + b_2_3·b_4_6·b_6_7
       + b_2_3·b_4_62·b_1_12 + b_2_3·b_4_5·b_1_1·b_5_0 + b_2_3·b_4_5·b_1_13·b_3_0
       + b_2_3·b_4_5·b_6_7 + b_2_3·b_4_5·b_4_6·b_1_12 + b_2_32·b_6_7·b_1_12
       + b_2_32·b_4_5·b_1_1·b_3_0 + b_2_32·b_4_5·b_1_14 + b_2_33·b_1_13·b_3_0
       + b_2_3·c_8_6·b_1_12
  44. b_4_6·b_3_0·b_5_0 + b_4_6·b_1_1·b_7_0 + b_4_6·b_8_7 + b_4_6·b_6_7·b_1_12
       + b_4_62·b_1_1·b_3_0 + b_4_62·b_1_0·b_3_0 + b_4_62·b_1_04 + b_4_63
       + b_4_5·b_6_7·b_1_12 + b_4_5·b_4_6·b_1_04 + b_4_5·b_4_62
       + b_2_3·b_4_6·b_1_13·b_3_0 + b_2_3·b_4_62·b_1_12 + b_2_3·b_4_5·b_6_7
       + b_2_32·b_1_1·b_7_0 + b_2_32·b_4_6·b_1_1·b_3_0 + b_2_32·b_4_62
       + b_2_32·b_4_5·b_1_1·b_3_0 + b_2_33·b_1_1·b_5_0 + b_2_33·b_4_6·b_1_12
       + b_2_34·b_1_1·b_3_0 + b_2_34·b_4_6 + b_2_34·b_4_5
  45. b_6_72 + b_4_62·b_1_0·b_3_0 + b_4_62·b_1_04 + b_4_63 + b_4_5·b_4_6·b_1_0·b_3_0
       + b_4_5·b_4_62 + b_4_52·b_1_14 + b_4_52·b_1_04 + b_2_3·b_6_7·b_1_14
       + b_2_3·b_4_6·b_6_7 + b_2_3·b_4_5·b_1_16 + b_2_3·b_4_5·b_4_6·b_1_12
       + b_2_32·b_1_1·b_7_0 + b_2_32·b_1_15·b_3_0 + b_2_32·b_8_7
       + b_2_32·b_4_6·b_1_1·b_3_0 + b_2_32·b_4_52 + b_2_33·b_1_1·b_5_0
       + b_2_33·b_1_13·b_3_0 + b_2_33·b_4_5·b_1_12 + b_2_34·b_1_1·b_3_0 + b_2_34·b_4_6
       + b_2_34·b_4_5 + c_8_6·b_1_14 + b_2_32·c_8_6
  46. b_5_0·b_7_16 + b_2_3·b_4_6·b_1_13·b_3_0 + b_2_3·b_4_6·b_6_7 + b_2_3·b_4_62·b_1_12
       + b_2_3·b_4_5·b_1_13·b_3_0 + b_2_32·b_8_7 + b_2_32·b_4_6·b_1_1·b_3_0
       + b_2_32·b_4_62 + b_2_33·b_1_1·b_5_0 + b_2_33·b_1_13·b_3_0 + b_2_33·b_6_7
       + b_2_33·b_4_6·b_1_12 + b_2_34·b_4_6 + b_2_34·b_4_5
  47. b_5_1·b_7_0 + b_4_6·b_1_1·b_7_0 + b_4_6·b_1_0·b_7_0 + b_4_6·b_6_7·b_1_12
       + b_4_62·b_1_1·b_3_0 + b_4_62·b_1_0·b_3_0 + b_4_63 + b_4_5·b_3_0·b_5_0
       + b_4_5·b_1_1·b_7_0 + b_4_5·b_8_7 + b_4_5·b_4_6·b_1_04 + b_4_52·b_1_1·b_3_0
       + b_4_52·b_1_14 + b_4_52·b_1_04 + b_4_52·b_4_6 + b_2_3·b_6_7·b_1_1·b_3_0
       + b_2_3·b_6_7·b_1_14 + b_2_3·b_4_6·b_1_13·b_3_0 + b_2_3·b_4_6·b_6_7
       + b_2_3·b_4_62·b_1_12 + b_2_3·b_4_5·b_1_13·b_3_0 + b_2_3·b_4_5·b_1_16
       + b_2_3·b_4_5·b_4_6·b_1_12 + b_2_32·b_1_1·b_7_0 + b_2_32·b_1_15·b_3_0
       + b_2_32·b_6_7·b_1_12 + b_2_32·b_4_6·b_1_1·b_3_0 + b_2_32·b_4_5·b_1_1·b_3_0
       + b_2_32·b_4_5·b_1_14 + b_2_32·b_4_52 + b_2_33·b_1_1·b_5_0
       + b_2_33·b_1_13·b_3_0 + b_2_33·b_4_6·b_1_12 + b_2_34·b_1_1·b_3_0 + b_2_34·b_4_6
       + b_2_34·b_4_5 + c_8_6·b_1_1·b_3_0 + c_8_6·b_1_14 + b_2_3·c_8_6·b_1_12
  48. b_5_1·b_7_16 + b_2_3·b_4_6·b_6_7 + b_2_3·b_4_62·b_1_12 + b_2_3·b_4_5·b_1_13·b_3_0
       + b_2_32·b_8_7 + b_2_32·b_4_6·b_1_1·b_3_0 + b_2_32·b_4_62 + b_2_33·b_1_1·b_5_0
       + b_2_33·b_1_13·b_3_0 + b_2_33·b_6_7 + b_2_33·b_4_6·b_1_12 + b_2_34·b_4_6
       + b_2_34·b_4_5
  49. b_6_7·b_7_0 + b_4_6·b_1_02·b_7_0 + b_4_62·b_5_1 + b_4_62·b_1_12·b_3_0
       + b_4_63·b_1_0 + b_4_5·b_1_0·b_1_1·b_7_0 + b_4_5·b_1_02·b_7_0 + b_4_5·b_4_6·b_5_1
       + b_4_5·b_4_6·b_1_12·b_3_0 + b_4_5·b_4_62·b_1_1 + b_4_5·b_4_62·b_1_0
       + b_4_52·b_1_12·b_3_0 + b_2_3·b_6_7·b_1_12·b_3_0 + b_2_3·b_4_6·b_6_7·b_1_1
       + b_2_3·b_4_5·b_1_14·b_3_0 + b_2_3·b_4_5·b_4_6·b_3_0 + b_2_3·b_4_52·b_3_0
       + b_2_32·b_4_6·b_5_0 + b_2_32·b_4_62·b_1_1 + b_2_32·b_4_5·b_5_0
       + b_2_32·b_4_5·b_1_12·b_3_0 + b_2_32·b_4_5·b_4_6·b_1_1 + b_2_32·b_4_52·b_1_1
       + b_2_33·b_7_0 + b_2_33·b_6_7·b_1_1 + b_2_33·b_4_6·b_3_0 + b_2_33·b_4_5·b_1_13
       + b_2_34·b_5_0 + b_2_34·b_1_12·b_3_0 + c_8_6·b_1_12·b_3_0 + b_4_6·c_8_6·b_1_1
       + b_2_3·c_8_6·b_3_5 + b_2_3·c_8_6·b_3_0 + b_2_3·c_8_6·b_1_13 + b_2_32·c_8_6·b_1_1
  50. b_6_7·b_7_16 + b_2_3·b_4_6·b_6_7·b_1_1 + b_2_33·b_7_16 + b_2_34·b_4_6·b_1_1
       + b_2_3·c_8_6·b_3_5
  51. b_8_7·b_5_0 + b_4_6·b_3_03 + b_4_6·b_1_04·b_5_0 + b_4_6·b_6_7·b_3_0
       + b_4_62·b_1_12·b_3_0 + b_4_5·b_3_03 + b_4_5·b_1_0·b_1_1·b_7_0
       + b_4_5·b_1_04·b_5_0 + b_4_5·b_6_7·b_3_0 + b_4_5·b_4_6·b_5_1 + b_4_5·b_4_6·b_5_0
       + b_4_52·b_4_6·b_1_0 + b_2_3·b_6_7·b_1_12·b_3_0 + b_2_3·b_4_6·b_7_0
       + b_2_3·b_4_62·b_3_0 + b_2_3·b_4_62·b_1_13 + b_2_3·b_4_5·b_1_14·b_3_0
       + b_2_3·b_4_5·b_4_6·b_3_0 + b_2_3·b_4_52·b_3_0 + b_2_3·b_4_52·b_1_13
       + b_2_32·b_6_7·b_1_13 + b_2_32·b_4_6·b_1_12·b_3_0 + b_2_32·b_4_62·b_1_1
       + b_2_32·b_4_5·b_5_0 + b_2_32·b_4_5·b_1_12·b_3_0 + b_2_32·b_4_5·b_1_15
       + b_2_33·b_1_14·b_3_0 + b_2_33·b_4_6·b_3_0 + b_2_33·b_4_6·b_1_13
       + b_2_33·b_4_5·b_1_13 + b_2_34·b_5_0 + b_2_34·b_4_5·b_1_1 + b_4_6·c_8_6·b_1_1
       + b_4_6·c_8_6·b_1_0 + b_2_3·c_8_6·b_1_13 + b_2_32·c_8_6·b_1_1
  52. b_8_7·b_5_1 + b_4_6·b_6_7·b_3_0 + b_4_62·b_5_1 + b_4_62·b_5_0 + b_4_62·b_1_05
       + b_4_5·b_4_6·b_5_1 + b_4_5·b_4_6·b_1_12·b_3_0 + b_4_5·b_4_6·b_1_05
       + b_4_5·b_4_62·b_1_0 + b_2_3·b_6_7·b_1_15 + b_2_3·b_4_6·b_7_0 + b_2_3·b_4_5·b_7_0
       + b_2_3·b_4_5·b_1_14·b_3_0 + b_2_3·b_4_5·b_1_17 + b_2_3·b_4_5·b_4_6·b_3_0
       + b_2_3·b_4_5·b_4_6·b_1_13 + b_2_3·b_4_52·b_3_0 + b_2_32·b_1_16·b_3_0
       + b_2_32·b_6_7·b_1_13 + b_2_32·b_4_6·b_1_12·b_3_0 + b_2_32·b_4_5·b_1_12·b_3_0
       + b_2_32·b_4_5·b_1_15 + b_2_33·b_1_14·b_3_0 + b_2_33·b_4_6·b_3_0
       + b_2_33·b_4_5·b_1_13 + b_2_34·b_5_0 + b_2_34·b_1_12·b_3_0 + b_2_34·b_4_6·b_1_1
       + b_4_6·c_8_6·b_1_1 + b_2_3·c_8_6·b_1_13 + b_2_32·c_8_6·b_1_1
  53. b_6_7·b_8_7 + b_4_6·b_6_7·b_1_1·b_3_0 + b_4_62·b_1_0·b_5_0 + b_4_62·b_1_06
       + b_4_62·b_6_7 + b_4_63·b_1_12 + b_4_63·b_1_02 + b_4_5·b_6_7·b_1_1·b_3_0
       + b_4_5·b_4_6·b_1_0·b_5_0 + b_4_52·b_1_06 + b_4_52·b_4_6·b_1_02
       + b_2_3·b_6_7·b_1_13·b_3_0 + b_2_3·b_4_6·b_1_1·b_7_0 + b_2_3·b_4_5·b_4_62
       + b_2_32·b_6_7·b_1_1·b_3_0 + b_2_32·b_6_7·b_1_14 + b_2_32·b_4_6·b_1_13·b_3_0
       + b_2_32·b_4_6·b_6_7 + b_2_32·b_4_5·b_1_16 + b_2_32·b_4_5·b_6_7
       + b_2_33·b_1_15·b_3_0 + b_2_33·b_8_7 + b_2_33·b_6_7·b_1_12
       + b_2_33·b_4_5·b_1_14 + b_2_33·b_4_5·b_4_6 + b_2_33·b_4_52
       + b_2_34·b_4_6·b_1_12 + b_2_34·b_4_5·b_1_12 + b_4_6·c_8_6·b_1_12
       + b_2_3·c_8_6·b_1_14 + b_2_3·b_4_6·c_8_6 + b_2_32·c_8_6·b_1_12 + b_2_33·c_8_6
  54. b_7_02 + b_1_02·b_5_0·b_7_0 + b_8_7·b_3_02 + b_4_6·b_3_0·b_7_0
       + b_4_6·b_1_05·b_5_0 + b_4_62·b_1_13·b_3_0 + b_4_62·b_1_0·b_5_0 + b_4_62·b_6_7
       + b_4_63·b_1_12 + b_4_5·b_1_05·b_5_0 + b_4_5·b_4_6·b_3_02 + b_4_5·b_4_6·b_1_06
       + b_4_5·b_4_62·b_1_02 + b_4_52·b_3_02 + b_4_52·b_1_1·b_5_0
       + b_4_52·b_1_13·b_3_0 + b_4_52·b_1_0·b_5_1 + b_4_52·b_1_06
       + b_4_52·b_4_6·b_1_12 + b_4_53·b_1_12 + b_4_53·b_1_0·b_1_1 + b_4_53·b_1_02
       + b_2_3·b_6_7·b_1_13·b_3_0 + b_2_3·b_4_6·b_1_1·b_7_0 + b_2_3·b_4_63
       + b_2_3·b_4_5·b_1_15·b_3_0 + b_2_3·b_4_5·b_6_7·b_1_12 + b_2_3·b_4_52·b_1_14
       + b_2_3·b_4_53 + b_2_32·b_3_5·b_7_0 + b_2_32·b_6_7·b_1_1·b_3_0
       + b_2_32·b_6_7·b_1_14 + b_2_32·b_4_5·b_1_13·b_3_0 + b_2_32·b_4_5·b_1_16
       + b_2_32·b_4_5·b_6_7 + b_2_33·b_1_15·b_3_0 + b_2_33·b_8_7 + b_2_33·b_6_7·b_1_12
       + b_2_33·b_4_62 + b_2_33·b_4_5·b_1_1·b_3_0 + b_2_33·b_4_5·b_1_14
       + b_2_33·b_4_52 + b_2_34·b_1_1·b_5_0 + b_2_34·b_6_7 + b_2_35·b_1_1·b_3_0
       + c_8_6·b_3_52 + c_8_6·b_3_02 + c_8_6·b_1_06 + b_4_6·c_8_6·b_1_12
       + b_4_5·c_8_6·b_1_12 + b_4_5·c_8_6·b_1_02 + b_2_32·c_8_6·b_1_12
  55. b_7_0·b_7_16 + b_2_3·b_4_6·b_1_1·b_7_0 + b_2_32·b_3_5·b_7_0 + c_8_6·b_3_52
  56. b_7_162 + b_2_32·b_3_5·b_7_0 + b_2_32·b_4_62·b_1_12 + c_8_6·b_3_52
  57. b_8_7·b_7_16 + b_2_3·b_4_62·b_5_0 + b_2_32·b_4_6·b_6_7·b_1_1
       + b_2_32·b_4_62·b_1_13 + b_2_32·b_4_5·b_1_14·b_3_0 + b_2_32·b_4_5·b_4_6·b_3_0
       + b_2_34·b_7_16 + b_2_34·b_6_7·b_1_1 + b_2_34·b_4_6·b_1_13 + b_2_34·b_4_5·b_3_0
       + b_2_35·b_5_0 + b_2_35·b_1_12·b_3_0 + b_2_35·b_4_6·b_1_1 + b_2_35·b_4_5·b_1_1
       + b_2_32·c_8_6·b_3_5
  58. b_3_0·b_5_0·b_7_0 + b_8_7·b_7_0 + b_4_6·b_1_04·b_7_0 + b_4_6·b_6_7·b_1_12·b_3_0
       + b_4_62·b_1_0·b_3_02 + b_4_63·b_3_0 + b_4_5·b_1_04·b_7_0
       + b_4_5·b_6_7·b_1_12·b_3_0 + b_4_5·b_4_6·b_7_0 + b_4_5·b_4_6·b_1_0·b_3_02
       + b_4_5·b_4_6·b_6_7·b_1_1 + b_4_5·b_4_62·b_3_0 + b_4_5·b_4_62·b_1_13
       + b_4_52·b_6_7·b_1_1 + b_4_53·b_1_13 + b_2_3·b_4_62·b_5_0 + b_2_3·b_4_63·b_1_1
       + b_2_3·b_4_5·b_4_6·b_5_0 + b_2_3·b_4_5·b_4_6·b_1_12·b_3_0 + b_2_3·b_4_53·b_1_0
       + b_2_32·b_4_6·b_6_7·b_1_1 + b_2_32·b_4_62·b_3_0 + b_2_32·b_4_5·b_7_0
       + b_2_32·b_4_5·b_1_14·b_3_0 + b_2_32·b_4_5·b_4_6·b_3_0
       + b_2_32·b_4_5·b_4_6·b_1_13 + b_2_33·b_6_7·b_3_0 + b_2_33·b_6_7·b_1_13
       + b_2_33·b_4_5·b_5_0 + b_2_33·b_4_5·b_1_12·b_3_0 + b_2_33·b_4_5·b_1_15
       + b_2_33·b_4_5·b_4_6·b_1_1 + b_2_34·b_7_16 + b_2_34·b_1_14·b_3_0
       + b_2_34·b_4_6·b_1_13 + b_2_34·b_4_5·b_3_0 + b_2_34·b_4_5·b_1_13 + b_2_35·b_5_0
       + b_2_35·b_4_6·b_1_1 + b_2_35·b_4_5·b_1_1 + b_6_7·c_8_6·b_1_1 + b_2_3·c_8_6·b_5_1
       + b_2_3·b_4_6·c_8_6·b_1_1 + b_2_3·b_4_5·c_8_6·b_1_0 + b_2_32·c_8_6·b_3_5
       + b_2_32·c_8_6·b_3_0 + b_2_32·c_8_6·b_1_13 + b_2_33·c_8_6·b_1_1
  59. b_8_72 + b_4_6·b_3_04 + b_4_62·b_1_08 + b_4_63·b_1_0·b_3_0 + b_4_5·b_3_04
       + b_4_5·b_4_62·b_1_0·b_3_0 + b_4_52·b_1_08 + b_4_52·b_4_6·b_1_0·b_3_0
       + b_2_3·b_4_62·b_1_13·b_3_0 + b_2_3·b_4_62·b_6_7 + b_2_3·b_4_63·b_1_12
       + b_2_3·b_4_5·b_4_6·b_6_7 + b_2_3·b_4_5·b_4_62·b_1_12
       + b_2_3·b_4_52·b_1_13·b_3_0 + b_2_32·b_6_7·b_1_16 + b_2_32·b_4_6·b_1_1·b_7_0
       + b_2_32·b_4_6·b_6_7·b_1_12 + b_2_32·b_4_5·b_1_1·b_7_0
       + b_2_32·b_4_5·b_1_15·b_3_0 + b_2_32·b_4_5·b_1_18 + b_2_32·b_4_5·b_8_7
       + b_2_32·b_4_5·b_4_62 + b_2_32·b_4_52·b_1_1·b_3_0 + b_2_32·b_4_52·b_4_6
       + b_2_32·b_4_53 + b_2_33·b_1_17·b_3_0 + b_2_33·b_6_7·b_1_1·b_3_0
       + b_2_33·b_4_6·b_6_7 + b_2_33·b_4_62·b_1_12 + b_2_33·b_4_5·b_1_1·b_5_0
       + b_2_33·b_4_5·b_1_13·b_3_0 + b_2_33·b_4_5·b_6_7 + b_2_34·b_8_7
       + b_2_34·b_4_6·b_1_1·b_3_0 + b_2_34·b_4_62 + b_2_34·b_4_5·b_1_1·b_3_0
       + b_2_34·b_4_5·b_1_14 + b_2_34·b_4_5·b_4_6 + b_4_6·c_8_6·b_1_0·b_3_0
       + b_2_3·b_4_6·c_8_6·b_1_12 + b_2_32·c_8_6·b_1_1·b_3_0 + b_2_32·c_8_6·b_1_14
       + b_2_32·b_4_6·c_8_6 + b_2_32·b_4_5·c_8_6 + b_2_34·c_8_6

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 16 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. b_3_0·b_5_0 + b_1_1·b_7_0 + b_1_18 + b_1_08 + b_4_62 + b_4_5·b_1_1·b_3_0 + b_4_5·b_4_6
         + b_4_52 + b_2_3·b_1_1·b_5_0 + b_2_3·b_6_7 + b_2_3·b_4_6·b_1_12
         + b_2_3·b_4_5·b_1_12 + b_2_32·b_4_6 + b_2_32·b_4_5 + b_2_34 + c_8_6, an element of degree 8
    2. b_3_54 + b_3_04 + b_4_6·b_1_03·b_5_0 + b_4_6·b_6_7·b_1_12 + b_4_62·b_1_1·b_3_0
         + b_4_62·b_1_0·b_3_0 + b_4_5·b_3_0·b_5_0 + b_4_5·b_1_1·b_7_0 + b_4_5·b_6_7·b_1_12
         + b_4_5·b_4_6·b_1_1·b_3_0 + b_4_5·b_4_6·b_1_0·b_3_0 + b_4_5·b_4_62
         + b_4_52·b_1_1·b_3_0 + b_4_52·b_1_04 + b_4_52·b_4_6 + b_2_3·b_4_6·b_1_13·b_3_0
         + b_2_3·b_4_62·b_1_12 + b_2_3·b_4_5·b_1_13·b_3_0 + b_2_3·b_4_5·b_6_7
         + b_2_32·b_8_7 + b_2_32·b_6_7·b_1_12 + b_2_32·b_4_6·b_1_1·b_3_0
         + b_2_32·b_4_5·b_1_1·b_3_0 + b_2_32·b_4_5·b_1_14 + b_2_34·b_1_14
         + c_8_6·b_1_0·b_3_0 + b_4_6·c_8_6 + b_4_5·c_8_6 + b_2_32·c_8_6, an element of degree 12
    3. b_8_7·b_3_02 + b_4_62·b_3_02 + b_4_62·b_1_0·b_5_0 + b_4_62·b_1_06
         + b_4_63·b_1_02 + b_4_5·b_4_6·b_3_02 + b_4_5·b_4_6·b_1_06
         + b_4_52·b_4_6·b_1_02 + b_2_3·b_6_7·b_1_13·b_3_0 + b_2_3·b_4_6·b_1_1·b_7_0
         + b_2_3·b_4_6·b_6_7·b_1_12 + b_2_3·b_4_62·b_1_1·b_3_0 + b_2_3·b_4_63
         + b_2_3·b_4_5·b_1_1·b_7_0 + b_2_3·b_4_5·b_1_15·b_3_0 + b_2_3·b_4_5·b_8_7
         + b_2_3·b_4_5·b_6_7·b_1_12 + b_2_3·b_4_52·b_1_1·b_3_0 + b_2_3·b_4_52·b_4_6
         + b_2_32·b_3_5·b_7_0 + b_2_32·b_6_7·b_1_1·b_3_0 + b_2_32·b_4_6·b_1_13·b_3_0
         + b_2_32·b_4_5·b_4_6·b_1_12 + b_2_32·b_4_52·b_1_12 + b_2_33·b_1_1·b_7_0
         + b_2_33·b_8_7 + b_2_33·b_6_7·b_1_12 + b_2_33·b_4_5·b_1_14 + b_2_34·b_1_1·b_5_0
         + b_2_34·b_6_7 + b_2_34·b_4_5·b_1_12 + b_2_35·b_1_1·b_3_0 + c_8_6·b_3_52
         + c_8_6·b_3_02 + c_8_6·b_1_1·b_5_0 + c_8_6·b_1_0·b_5_0 + c_8_6·b_1_06
         + b_4_6·c_8_6·b_1_02 + b_4_5·c_8_6·b_1_12 + b_4_5·c_8_6·b_1_02
         + b_2_3·c_8_6·b_1_1·b_3_0 + b_2_3·c_8_6·b_1_14, an element of degree 14
    4. b_1_1 + b_1_0, an element of degree 1
  • A Duflot regular sequence is given by c_8_6.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 17, 29, 31].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. b_3_0·b_5_0 + b_1_1·b_7_0 + b_1_18 + b_1_08 + b_4_62 + b_4_5·b_1_1·b_3_0 + b_4_5·b_4_6
         + b_4_52 + b_2_3·b_1_1·b_5_0 + b_2_3·b_6_7 + b_2_3·b_4_6·b_1_12
         + b_2_3·b_4_5·b_1_12 + b_2_32·b_4_6 + b_2_32·b_4_5 + b_2_34 + c_8_6, an element of degree 8
    2. b_3_54 + b_3_04 + b_4_6·b_1_03·b_5_0 + b_4_6·b_6_7·b_1_12 + b_4_62·b_1_1·b_3_0
         + b_4_62·b_1_0·b_3_0 + b_4_5·b_3_0·b_5_0 + b_4_5·b_1_1·b_7_0 + b_4_5·b_6_7·b_1_12
         + b_4_5·b_4_6·b_1_1·b_3_0 + b_4_5·b_4_6·b_1_0·b_3_0 + b_4_5·b_4_62
         + b_4_52·b_1_1·b_3_0 + b_4_52·b_1_04 + b_4_52·b_4_6 + b_2_3·b_4_6·b_1_13·b_3_0
         + b_2_3·b_4_62·b_1_12 + b_2_3·b_4_5·b_1_13·b_3_0 + b_2_3·b_4_5·b_6_7
         + b_2_32·b_8_7 + b_2_32·b_6_7·b_1_12 + b_2_32·b_4_6·b_1_1·b_3_0
         + b_2_32·b_4_5·b_1_1·b_3_0 + b_2_32·b_4_5·b_1_14 + b_2_34·b_1_14
         + c_8_6·b_1_0·b_3_0 + b_4_6·c_8_6 + b_4_5·c_8_6 + b_2_32·c_8_6, an element of degree 12
    3. b_3_5 + b_3_0, an element of degree 3
    4. b_1_1 + b_1_0, an element of degree 1
  • We found that there exists some HSOP over a finite extension field, in degrees 8,3,1,4.

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(Syl2(M22); GF(2))

  1. b_1_1 → b_1_1
  2. b_1_0 → b_1_2
  3. b_2_3 → b_1_02 + b_2_4
  4. b_3_5 → b_2_4·b_1_0
  5. b_3_0 → b_3_8 + b_2_5·b_1_2 + b_2_5·b_1_0
  6. b_4_6 → b_4_9 + b_2_5·b_1_22 + b_2_5·b_1_02
  7. b_4_5 → b_4_13 + b_2_5·b_1_02 + b_2_52
  8. b_5_1 → b_5_20 + b_4_10·b_1_2 + b_2_5·b_1_23 + b_2_52·b_1_0 + b_2_4·b_3_8
  9. b_5_0 → b_5_21 + b_5_17 + b_4_10·b_1_2 + b_2_52·b_1_2 + b_2_52·b_1_0
  10. b_6_7 → b_6_30 + b_4_10·b_1_22 + b_2_5·b_1_24 + b_2_5·b_4_10 + b_2_52·b_1_22
  11. b_7_16 → b_2_4·b_5_17
  12. b_7_0 → b_7_41 + b_4_10·b_3_8 + b_4_10·b_1_23 + b_2_5·b_5_21 + b_2_5·b_1_25
       + b_2_5·b_4_9·b_1_2
  13. b_8_7 → b_8_52 + b_4_10·b_1_24 + b_4_9·b_4_10 + b_2_5·b_1_26 + b_2_5·b_4_10·b_1_22
       + b_2_52·b_1_24 + b_2_52·b_4_13 + b_2_42·b_4_9
  14. c_8_6 → b_8_54 + b_2_5·b_6_31 + b_2_5·b_4_9·b_1_22 + b_2_52·b_1_24 + b_2_52·b_4_13
       + b_2_52·b_4_9 + b_2_42·b_4_13 + b_2_42·b_4_9 + c_8_55

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

  1. b_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_3 → 0, an element of degree 2
  4. b_3_5 → 0, an element of degree 3
  5. b_3_0 → 0, an element of degree 3
  6. b_4_6 → 0, an element of degree 4
  7. b_4_5 → 0, an element of degree 4
  8. b_5_1 → 0, an element of degree 5
  9. b_5_0 → 0, an element of degree 5
  10. b_6_7 → 0, an element of degree 6
  11. b_7_16 → 0, an element of degree 7
  12. b_7_0 → 0, an element of degree 7
  13. b_8_7 → 0, an element of degree 8
  14. c_8_6 → c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. b_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_3 → c_1_22 + c_1_1·c_1_2 + c_1_12, an element of degree 2
  4. b_3_5 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  5. b_3_0 → 0, an element of degree 3
  6. b_4_6 → 0, an element of degree 4
  7. b_4_5 → 0, an element of degree 4
  8. b_5_1 → 0, an element of degree 5
  9. b_5_0 → 0, an element of degree 5
  10. b_6_7 → c_1_0·c_1_1·c_1_24 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_24
       + c_1_02·c_1_12·c_1_22 + c_1_02·c_1_14 + c_1_04·c_1_22 + c_1_04·c_1_1·c_1_2
       + c_1_04·c_1_12, an element of degree 6
  11. b_7_16 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
  12. b_7_0 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
  13. b_8_7 → c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24
       + c_1_0·c_1_14·c_1_23 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_13·c_1_23
       + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14, an element of degree 8
  14. c_8_6 → c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25 + c_1_0·c_1_13·c_1_24
       + c_1_0·c_1_14·c_1_23 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2
       + c_1_02·c_1_16 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup

  1. b_1_1 → 0, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_3 → c_1_12, an element of degree 2
  4. b_3_5 → 0, an element of degree 3
  5. b_3_0 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
  6. b_4_6 → c_1_12·c_1_22 + c_1_13·c_1_2, an element of degree 4
  7. b_4_5 → c_1_24 + c_1_13·c_1_2, an element of degree 4
  8. b_5_1 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  9. b_5_0 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
  10. b_6_7 → c_1_0·c_1_13·c_1_22 + c_1_0·c_1_14·c_1_2 + c_1_02·c_1_12·c_1_22
       + c_1_02·c_1_13·c_1_2 + c_1_02·c_1_14 + c_1_04·c_1_12, an element of degree 6
  11. b_7_16 → 0, an element of degree 7
  12. b_7_0 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_2 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2, an element of degree 7
  13. b_8_7 → c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16 + c_1_04·c_1_12·c_1_22
       + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14, an element of degree 8
  14. c_8_6 → c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_1 → c_1_1, an element of degree 1
  2. b_1_0 → 0, an element of degree 1
  3. b_2_3 → c_1_22 + c_1_1·c_1_2, an element of degree 2
  4. b_3_5 → 0, an element of degree 3
  5. b_3_0 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  6. b_4_6 → c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_13·c_1_3 + c_1_0·c_1_1·c_1_22 + c_1_0·c_1_12·c_1_2, an element of degree 4
  7. b_4_5 → c_1_34 + c_1_23·c_1_3 + c_1_12·c_1_2·c_1_3 + c_1_13·c_1_3 + c_1_0·c_1_1·c_1_22
       + c_1_0·c_1_12·c_1_2, an element of degree 4
  8. b_5_1 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_3
       + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_0·c_1_14
       + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_1, an element of degree 5
  9. b_5_0 → c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_1·c_1_22·c_1_32 + c_1_1·c_1_23·c_1_3
       + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_0·c_1_12·c_1_22
       + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2
       + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_6_7 → c_1_1·c_1_22·c_1_33 + c_1_1·c_1_23·c_1_32 + c_1_12·c_1_2·c_1_33
       + c_1_13·c_1_2·c_1_32 + c_1_13·c_1_22·c_1_3 + c_1_14·c_1_32
       + c_1_14·c_1_2·c_1_3 + c_1_15·c_1_3 + c_1_0·c_1_23·c_1_32 + c_1_0·c_1_24·c_1_3
       + c_1_0·c_1_12·c_1_2·c_1_32 + c_1_0·c_1_12·c_1_22·c_1_3 + c_1_0·c_1_13·c_1_32
       + c_1_0·c_1_14·c_1_3 + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_23·c_1_3
       + c_1_02·c_1_24 + c_1_02·c_1_1·c_1_2·c_1_32 + c_1_02·c_1_1·c_1_22·c_1_3
       + c_1_02·c_1_12·c_1_32 + c_1_02·c_1_13·c_1_3 + c_1_02·c_1_13·c_1_2
       + c_1_02·c_1_14 + c_1_03·c_1_1·c_1_22 + c_1_03·c_1_12·c_1_2 + c_1_04·c_1_22
       + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
  11. b_7_16 → c_1_1·c_1_24·c_1_32 + c_1_1·c_1_25·c_1_3 + c_1_13·c_1_23·c_1_3
       + c_1_14·c_1_2·c_1_32 + c_1_14·c_1_22·c_1_3 + c_1_15·c_1_2·c_1_3
       + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22, an element of degree 7
  12. b_7_0 → c_1_1·c_1_36 + c_1_1·c_1_22·c_1_34 + c_1_1·c_1_24·c_1_32 + c_1_1·c_1_25·c_1_3
       + c_1_12·c_1_35 + c_1_12·c_1_22·c_1_33 + c_1_13·c_1_2·c_1_33
       + c_1_13·c_1_23·c_1_3 + c_1_14·c_1_33 + c_1_14·c_1_22·c_1_3 + c_1_15·c_1_32
       + c_1_15·c_1_2·c_1_3 + c_1_0·c_1_22·c_1_34 + c_1_0·c_1_24·c_1_32
       + c_1_0·c_1_1·c_1_23·c_1_32 + c_1_0·c_1_1·c_1_24·c_1_3 + c_1_0·c_1_12·c_1_34
       + c_1_0·c_1_12·c_1_22·c_1_32 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_2·c_1_3
       + c_1_0·c_1_14·c_1_22 + c_1_0·c_1_15·c_1_3 + c_1_02·c_1_2·c_1_34
       + c_1_02·c_1_24·c_1_3 + c_1_02·c_1_1·c_1_34 + c_1_02·c_1_1·c_1_23·c_1_3
       + c_1_02·c_1_1·c_1_24 + c_1_02·c_1_13·c_1_2·c_1_3 + c_1_02·c_1_14·c_1_3
       + c_1_02·c_1_14·c_1_2 + c_1_03·c_1_12·c_1_22 + c_1_03·c_1_13·c_1_2
       + c_1_03·c_1_14 + c_1_04·c_1_2·c_1_32 + c_1_04·c_1_22·c_1_3 + c_1_04·c_1_13
       + c_1_05·c_1_12 + c_1_06·c_1_1, an element of degree 7
  13. b_8_7 → c_1_1·c_1_2·c_1_36 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_24·c_1_33
       + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_2·c_1_35 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_25·c_1_3 + c_1_13·c_1_2·c_1_34 + c_1_13·c_1_22·c_1_33
       + c_1_13·c_1_24·c_1_3 + c_1_14·c_1_2·c_1_33 + c_1_15·c_1_22·c_1_3
       + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_26·c_1_3 + c_1_0·c_1_1·c_1_24·c_1_32
       + c_1_0·c_1_1·c_1_25·c_1_3 + c_1_0·c_1_12·c_1_2·c_1_34
       + c_1_0·c_1_12·c_1_23·c_1_32 + c_1_0·c_1_13·c_1_22·c_1_32
       + c_1_0·c_1_13·c_1_23·c_1_3 + c_1_0·c_1_14·c_1_2·c_1_32
       + c_1_0·c_1_14·c_1_22·c_1_3 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2
       + c_1_02·c_1_22·c_1_34 + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_26
       + c_1_02·c_1_1·c_1_2·c_1_34 + c_1_02·c_1_1·c_1_24·c_1_3 + c_1_02·c_1_1·c_1_25
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_12·c_1_23·c_1_3
       + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_32 + c_1_02·c_1_14·c_1_22
       + c_1_02·c_1_15·c_1_3 + c_1_03·c_1_13·c_1_22 + c_1_03·c_1_14·c_1_2
       + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_23·c_1_3 + c_1_04·c_1_24
       + c_1_04·c_1_1·c_1_2·c_1_32 + c_1_04·c_1_1·c_1_22·c_1_3
       + c_1_04·c_1_12·c_1_32 + c_1_04·c_1_13·c_1_3 + c_1_04·c_1_13·c_1_2
       + c_1_05·c_1_1·c_1_22 + c_1_05·c_1_12·c_1_2, an element of degree 8
  14. c_8_6 → c_1_1·c_1_24·c_1_33 + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_24·c_1_32 + c_1_12·c_1_25·c_1_3 + c_1_13·c_1_2·c_1_34
       + c_1_13·c_1_22·c_1_33 + c_1_13·c_1_24·c_1_3 + c_1_14·c_1_23·c_1_3
       + c_1_15·c_1_2·c_1_32 + c_1_0·c_1_25·c_1_32 + c_1_0·c_1_26·c_1_3
       + c_1_0·c_1_1·c_1_22·c_1_34 + c_1_0·c_1_1·c_1_25·c_1_3
       + c_1_0·c_1_12·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_23·c_1_32
       + c_1_0·c_1_13·c_1_23·c_1_3 + c_1_0·c_1_15·c_1_2·c_1_3 + c_1_02·c_1_22·c_1_34
       + c_1_02·c_1_25·c_1_3 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_2·c_1_34
       + c_1_02·c_1_1·c_1_24·c_1_3 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_34
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_12·c_1_23·c_1_3
       + c_1_02·c_1_13·c_1_2·c_1_32 + c_1_02·c_1_13·c_1_22·c_1_3
       + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_32 + c_1_02·c_1_15·c_1_2
       + c_1_03·c_1_1·c_1_24 + c_1_03·c_1_13·c_1_22 + c_1_04·c_1_34
       + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_1·c_1_2·c_1_32
       + c_1_04·c_1_1·c_1_22·c_1_3 + c_1_04·c_1_12·c_1_32
       + c_1_04·c_1_12·c_1_2·c_1_3 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2
       + c_1_04·c_1_14 + c_1_08, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 4 in a Sylow subgroup

  1. b_1_1 → 0, an element of degree 1
  2. b_1_0 → c_1_1, an element of degree 1
  3. b_2_3 → 0, an element of degree 2
  4. b_3_5 → 0, an element of degree 3
  5. b_3_0 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_1·c_1_22 + c_1_12·c_1_3
       + c_1_12·c_1_2 + c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
  6. b_4_6 → c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3 + c_1_12·c_1_32 + c_1_12·c_1_22
       + c_1_13·c_1_3 + c_1_13·c_1_2 + c_1_0·c_1_13 + c_1_02·c_1_12, an element of degree 4
  7. b_4_5 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_2·c_1_32 + c_1_1·c_1_22·c_1_3
       + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_12·c_1_22, an element of degree 4
  8. b_5_1 → c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_13·c_1_32 + c_1_13·c_1_22
       + c_1_14·c_1_3 + c_1_14·c_1_2 + c_1_0·c_1_14 + c_1_02·c_1_13, an element of degree 5
  9. b_5_0 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_34 + c_1_1·c_1_24
       + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_13·c_1_32 + c_1_13·c_1_22
       + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
  10. b_6_7 → c_1_12·c_1_34 + c_1_12·c_1_22·c_1_32 + c_1_12·c_1_24 + c_1_14·c_1_2·c_1_3
       + c_1_15·c_1_3 + c_1_15·c_1_2 + c_1_0·c_1_15 + c_1_02·c_1_14, an element of degree 6
  11. b_7_16 → 0, an element of degree 7
  12. b_7_0 → c_1_2·c_1_36 + c_1_22·c_1_35 + c_1_23·c_1_34 + c_1_24·c_1_33
       + c_1_25·c_1_32 + c_1_26·c_1_3 + c_1_1·c_1_24·c_1_32 + c_1_12·c_1_2·c_1_34
       + c_1_14·c_1_22·c_1_3 + c_1_15·c_1_32 + c_1_15·c_1_2·c_1_3 + c_1_15·c_1_22
       + c_1_16·c_1_3 + c_1_16·c_1_2 + c_1_0·c_1_22·c_1_34 + c_1_0·c_1_24·c_1_32
       + c_1_0·c_1_12·c_1_34 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_32
       + c_1_0·c_1_14·c_1_22 + c_1_0·c_1_16 + c_1_02·c_1_2·c_1_34
       + c_1_02·c_1_24·c_1_3 + c_1_02·c_1_1·c_1_34 + c_1_02·c_1_1·c_1_24
       + c_1_02·c_1_14·c_1_3 + c_1_02·c_1_14·c_1_2 + c_1_03·c_1_14
       + c_1_04·c_1_2·c_1_32 + c_1_04·c_1_22·c_1_3 + c_1_04·c_1_1·c_1_32
       + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_3 + c_1_04·c_1_12·c_1_2
       + c_1_05·c_1_12 + c_1_06·c_1_1, an element of degree 7
  13. b_8_7 → c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_25·c_1_33 + c_1_26·c_1_32
       + c_1_1·c_1_2·c_1_36 + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_2·c_1_35
       + c_1_12·c_1_25·c_1_3 + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33
       + c_1_14·c_1_22·c_1_32 + c_1_14·c_1_23·c_1_3 + c_1_14·c_1_24
       + c_1_15·c_1_2·c_1_32 + c_1_15·c_1_22·c_1_3 + c_1_16·c_1_2·c_1_3 + c_1_17·c_1_3
       + c_1_17·c_1_2 + c_1_0·c_1_12·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_24·c_1_3
       + c_1_0·c_1_13·c_1_22·c_1_32 + c_1_0·c_1_15·c_1_2·c_1_3 + c_1_0·c_1_17
       + c_1_02·c_1_1·c_1_2·c_1_34 + c_1_02·c_1_1·c_1_24·c_1_3
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_13·c_1_2·c_1_32
       + c_1_02·c_1_13·c_1_22·c_1_3 + c_1_02·c_1_14·c_1_32
       + c_1_02·c_1_14·c_1_2·c_1_3 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_3
       + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_03·c_1_15
       + c_1_04·c_1_1·c_1_2·c_1_32 + c_1_04·c_1_1·c_1_22·c_1_3
       + c_1_04·c_1_12·c_1_32 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_3
       + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12, an element of degree 8
  14. c_8_6 → c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_25·c_1_33 + c_1_26·c_1_32
       + c_1_1·c_1_2·c_1_36 + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_2·c_1_35
       + c_1_12·c_1_22·c_1_34 + c_1_12·c_1_24·c_1_32 + c_1_12·c_1_25·c_1_3
       + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_23·c_1_3 + c_1_14·c_1_24
       + c_1_15·c_1_2·c_1_32 + c_1_15·c_1_22·c_1_3 + c_1_16·c_1_32 + c_1_16·c_1_22
       + c_1_0·c_1_1·c_1_22·c_1_34 + c_1_0·c_1_1·c_1_24·c_1_32
       + c_1_0·c_1_13·c_1_22·c_1_32 + c_1_0·c_1_14·c_1_2·c_1_32
       + c_1_0·c_1_14·c_1_22·c_1_3 + c_1_0·c_1_15·c_1_2·c_1_3 + c_1_02·c_1_22·c_1_34
       + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_12·c_1_34 + c_1_02·c_1_12·c_1_24
       + c_1_02·c_1_13·c_1_2·c_1_32 + c_1_02·c_1_13·c_1_22·c_1_3
       + c_1_02·c_1_15·c_1_3 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_03·c_1_15
       + c_1_04·c_1_34 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_24
       + c_1_04·c_1_12·c_1_2·c_1_3 + c_1_04·c_1_13·c_1_3 + c_1_04·c_1_13·c_1_2
       + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12 + c_1_08, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps


Simon King
Department of Mathematics and Computer Science
Friedrich-Schiller-Universität Jena
07737 Jena
GERMANY
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46161
Fax: +49 (0)3641 9-46162
Office: Zi. 3529, Ernst-Abbe-Platz 2

Last change: 14.12.2010