Mod-5-Cohomology of Normalizer(McL,Centre(SylowSubgroup(McL,5))), a group of order 3000

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • Normalizer(McL,Centre(SylowSubgroup(McL,5))) is a group of order 3000.
  • The group order factors as 23 · 3 · 53.
  • The group is defined by Group([(1,11,103,200,109)(2,147,89,202,14)(3,163,127,230,183)(4,206,104,193,140)(5,265,224,149,216)(6,213,253,129,190)(7,75,71,85,232)(8,176,143,245,142)(9,271,122,17,12)(10,132,119,106,195)(13,46,115,257,65)(15,50,275,239,20)(16,99,218,269,87)(18,171,35,32,270)(19,244,29,24,146)(21,105,196,214,264)(22,172,226,150,31)(23,101,237,113,39)(25,79,259,78,182)(26,66,207,145,112)(27,167,102,72,177)(28,73,199,70,118)(30,263,250,135,124)(33,184,59,210,111)(34,238,57,131,164)(36,191,123,148,83)(37,273,261,156,208)(38,86,60,134,43)(40,91,272,217,62)(41,45,139,49,234)(42,194,157,68,92)(44,198,185,116,136)(47,266,247,170,52)(48,220,260,243,252)(51,186,179,256,262)(53,56,222,248,94)(54,98,241,84,219)(55,93,125,187,126)(58,128,212,76,162)(61,96,108,204,211)(63,233,154,160,178)(64,189,114,144,107)(67,180,215,205,137)(69,100,246,74,138)(77,82,201,209,274)(80,110,267,169,141)(81,166,242,197,175)(88,174,173,95,168)(90,120,203,165,251)(97,152,151,231,155)(117,268,159,158,192)(121,235,225,240,221)(130,229,258,254,228)(133,161,249,181,227)(153,255,223,236,188),(1,109,200,103,11)(2,14,202,89,147)(3,183,230,127,163)(4,140,193,104,206)(5,216,149,224,265)(6,190,129,253,213)(7,232,85,71,75)(8,142,245,143,176)(9,12,17,122,271)(10,195,106,119,132)(13,65,257,115,46)(15,20,239,275,50)(16,87,269,218,99)(18,270,32,35,171)(19,146,24,29,244)(21,264,214,196,105)(22,31,150,226,172)(23,39,113,237,101)(25,182,78,259,79)(26,112,145,207,66)(27,177,72,102,167)(28,118,70,199,73)(30,124,135,250,263)(33,111,210,59,184)(34,164,131,57,238)(36,83,148,123,191)(37,208,156,261,273)(38,43,134,60,86)(40,62,217,272,91)(41,234,49,139,45)(42,92,68,157,194)(44,136,116,185,198)(47,52,170,247,266)(48,252,243,260,220)(51,262,256,179,186)(53,94,248,222,56)(54,219,84,241,98)(55,126,187,125,93)(58,162,76,212,128)(61,211,204,108,96)(63,178,160,154,233)(64,107,144,114,189)(67,137,205,215,180)(69,138,74,246,100)(77,274,209,201,82)(80,141,169,267,110)(81,175,197,242,166)(88,168,95,173,174)(90,251,165,203,120)(97,155,231,151,152)(117,192,158,159,268)(121,221,240,225,235)(130,228,254,258,229)(133,227,181,249,161)(153,188,236,223,255),(2,205,71)(3,219,74)(4,255,20)(5,119,148)(6,101,95)(7,202,180)(8,12,120)(9,203,176)(10,191,149)(13,33,76)(14,215,75)(15,206,223)(16,247,199)(17,90,142)(18,262,152)(19,161,64)(21,136,258)(22,125,243)(23,173,190)(24,227,144)(25,197,273)(26,49,80)(27,53,164)(28,269,47)(29,181,114)(30,201,192)(31,93,260)(32,179,155)(34,167,56)(35,186,231)(36,224,195)(37,182,242)(38,108,91)(39,174,129)(40,43,96)(41,267,207)(42,63,121)(44,254,105)(45,169,145)(46,184,162)(48,226,126)(50,104,236)(51,151,171)(52,118,218)(54,138,163)(55,220,150)(57,72,248)(58,115,59)(60,211,217)(61,62,134)(65,111,212)(66,234,110)(67,232,89)(68,160,240)(69,127,98)(70,99,170)(73,87,266)(77,159,135)(78,166,208)(79,175,261)(81,156,259)(82,158,124)(83,265,106)(84,246,183)(85,147,137)(86,204,272)(88,253,113)(92,178,221)(94,131,177)(97,270,256)(100,230,241)(102,222,238)(107,146,133)(112,139,141)(116,229,264)(117,263,209)(122,251,245)(123,216,132)(128,257,210)(130,214,185)(140,153,239)(143,271,165)(154,225,157)(168,213,237)(172,187,252)(188,275,193)(189,244,249)(194,233,235)(196,198,228)(250,274,268),(2,34,205,167,71,56)(3,42,219,63,74,121)(4,141,255,112,20,139)(5,148,119)(6,229,101,264,95,116)(7,94,202,131,180,177)(8,159,12,135,120,77)(9,124,203,82,176,158)(10,149,191)(13,47,33,28,76,269)(14,164,215,27,75,53)(15,49,206,80,223,26)(16,115,247,59,199,58)(17,250,90,274,142,268)(18,55,262,220,152,150)(19,64,161)(21,168,136,213,258,237)(22,35,125,186,243,231)(23,214,173,185,190,130)(24,144,227)(25,272,197,86,273,204)(29,114,181)(30,165,201,143,192,271)(31,171,93,51,260,151)(32,187,179,252,155,172)(36,195,224)(37,108,182,91,242,38)(39,196,174,198,129,228)(40,166,43,208,96,78)(41,193,267,188,207,275)(44,253,254,113,105,88)(45,140,169,153,145,239)(46,266,184,73,162,87)(48,97,226,270,126,256)(50,234,104,110,236,66)(52,111,118,212,218,65)(54,233,138,235,163,194)(57,67,72,232,248,89)(60,261,211,79,217,175)(61,259,62,81,134,156)(68,241,160,100,240,230)(69,225,127,157,98,154)(70,128,99,257,170,210)(83,106,265)(84,178,246,221,183,92)(85,222,147,238,137,102)(107,133,146)(117,122,263,251,209,245)(123,132,216)(189,249,244),(2,52,167,212)(3,182,63,38)(4,165,112,192)(5,216)(6,179,264,172)(7,87,131,184)(8,234,135,236)(9,80,82,15)(10,106)(11,109)(12,110,77,50)(13,215,28,53)(14,47,27,76)(16,57,59,232)(17,267,274,275)(18,228,220,174)(19,146)(20,271,141,201)(21,22,213,186)(23,97,185,126)(24,244)(25,178,86,183)(26,158,206,203)(30,255,143,139)(31,253,51,105)(32,229,252,95)(33,75,269,164)(34,111,71,218)(35,258,243,168)(37,42,91,74)(39,152,198,55)(40,138,208,194)(41,250,188,142)(43,163,78,233)(44,93,113,151)(45,263,153,245)(46,180,73,94)(48,173,270,130)(49,124,223,176)(54,166,235,96)(56,65,205,118)(58,89,247,72)(60,230,79,160)(61,98,81,225)(62,69,156,157)(64,107)(66,159,104,120)(67,199,248,115)(68,217,100,261)(70,222,257,137)(83,191)(84,197,221,204)(85,99,238,210)(88,171,254,260)(90,207,268,193)(92,272,246,273)(101,155,116,187)(102,128,147,170)(103,200)(108,219,242,121)(117,140,251,145)(119,132)(122,169,209,239)(123,148)(125,237,231,136)(127,259,154,134)(129,262,196,150)(133,161)(144,189)(149,265)(162,202,266,177)(175,240,211,241)(190,256,214,226)(227,249),(2,86,212,178,167,25,52,183)(3,202,38,162,63,177,182,266)(4,256,192,190,112,226,165,214)(5,249,216,227)(6,207,172,90,264,193,179,268)(7,108,184,121,131,242,87,219)(8,228,236,18,135,174,234,220)(9,136,15,231,82,237,80,125)(10,19,106,146)(11,200,109,103)(12,198,50,152,77,39,110,55)(13,157,53,156,28,69,215,62)(14,134,76,154,27,259,47,127)(16,241,232,211,59,240,57,175)(17,116,275,155,274,101,267,187)(20,97,201,23,141,126,271,185)(21,206,186,158,213,26,22,203)(24,119,244,132)(29,195)(30,173,139,48,143,130,255,270)(31,251,105,140,51,117,253,145)(32,250,95,41,252,142,229,188)(33,225,164,81,269,98,75,61)(34,197,218,84,71,204,111,221)(35,124,168,49,243,176,258,223)(36,114)(37,73,74,180,91,46,42,94)(40,257,194,222,208,70,138,137)(43,128,233,102,78,170,163,147)(44,239,151,209,113,169,93,122)(45,260,245,254,153,171,263,88)(54,85,96,210,235,238,166,99)(56,273,118,246,205,272,65,92)(58,160,72,79,247,230,89,60)(64,83,107,191)(66,150,120,196,104,262,159,129)(67,217,115,68,248,261,199,100)(123,144,148,189)(133,149,161,265)(181,224),(1,4,6,45,256,231,150,130)(2,64,106,91,170,138,72,194)(3,7,38,5,178,16,259,131)(8,48,165,12,20,93,174,268)(9,275,55,95,159,245,252,90)(10,272,266,74,27,157,202,107)(11,193,190,49,262,152,226,258)(13,161,73,175,36,221,56,84)(14,114,195,62,247,100,177,92)(15,126,88,158,176,243,203,271)(17,50,125,168,117,143,220,120)(18,207,44,23,169,30,153,82)(19,59)(21,214,196,264)(22,229,200,140,253,139,186,151)(24,184,29,210)(25,57,127,75,134,216,233,99)(26,136,237,141,135,188,209,171)(28,166,191,225,53,54,46,227)(31,254,109,104,213,234,179,97)(32,66,185,101,110,124,223,201)(33,146,111,244)(34,163,85,43,224,63,269,79)(35,145,116,39,80,263,236,77)(37,212)(40,47,69,167,42,89,189,132)(41,51,155,172,228,103,206,129)(52,246,102,68,147,144,119,217)(58,208,162,273)(60,265,154,87,182,164,230,232)(61,205,204,215)(65,181,199,242,83,235,222,98)(67,211)(70,81,148,240,248,219,257,133)(71,86,149,160,218,78,238,183)(76,156,128,261)(94,241,115,249,118,197,123,121)(96,180,108,137)(112,198,113,267,250,255,274,270)(122,239,187,173,192,142,260,251)]).
  • 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

The computation was based on 23 stability conditions for H*(E125; GF(5)).

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*(E125; GF(5))

  1. a_4_0 → a_1_1·a_3_5 + a_1_1·a_3_4 + a_1_0·a_3_5 − 2·a_1_0·a_3_4
  2. a_5_0 → b_2_3·a_3_5 + 2·b_2_2·a_3_5 − 2·b_2_2·a_3_4 + 2·b_2_2·b_2_3·a_1_1 − 2·b_2_22·a_1_1
  3. a_7_0 → a_7_8 + 2·b_2_33·a_1_1 + 2·b_2_2·b_2_3·a_3_5 + b_2_2·b_2_32·a_1_1 − 2·b_2_22·a_3_5
       + 2·b_2_22·a_3_4 + 2·b_2_22·b_2_3·a_1_1 − b_2_23·a_1_1 + 2·b_2_23·a_1_0
  4. b_8_0 → b_8_9 − b_2_34 + b_2_2·b_2_33 − 2·b_2_22·b_2_32 − b_2_24
       + b_2_2·b_2_3·a_1_1·a_3_5 + b_2_22·a_1_1·a_3_5 + 2·b_2_22·a_1_0·a_3_4
  5. a_13_1 → b_2_24·b_2_32·a_1_1 − b_2_25·b_2_3·a_1_1 + b_2_3·c_10_12·a_1_1
       + 2·b_2_2·c_10_12·a_1_1 − 2·b_2_2·c_10_12·a_1_0
  6. b_14_0 → b_2_24·b_2_33 − b_2_25·b_2_32 + 2·b_2_35·a_1_1·a_3_5
       − 2·b_2_23·b_2_32·a_1_1·a_3_5 − 2·b_2_25·a_1_1·a_3_5 − b_2_25·a_1_0·a_3_5
       + b_2_25·a_1_0·a_3_4 + b_2_32·c_10_12 + 2·b_2_2·b_2_3·c_10_12 − 2·b_2_22·c_10_12
  7. a_15_1 → a_2_0·c_10_12·a_3_5
  8. a_16_1 → c_10_12·a_3_4·a_3_5 − 2·b_2_2·c_10_12·a_1_1·a_3_5 − 2·b_2_2·c_10_12·a_1_0·a_3_5
  9. a_23_1 → a_2_0·c_10_122·a_1_1
  10. a_24_1 → c_10_122·a_1_1·a_3_4 − c_10_122·a_1_0·a_3_5
  11. a_39_1 → b_2_215·b_2_33·a_3_5 − 2·b_2_216·b_2_32·a_3_5 + 2·b_2_216·b_2_33·a_1_1
       + b_2_217·b_2_3·a_3_5 + b_2_217·b_2_32·a_1_1 + 2·b_2_218·b_2_3·a_1_1
       + b_2_210·b_2_33·c_10_12·a_3_5 − 2·b_2_211·b_2_32·c_10_12·a_3_5
       + 2·b_2_211·b_2_33·c_10_12·a_1_1 − 2·b_2_212·b_2_3·c_10_12·a_3_5
       + b_2_212·b_2_32·c_10_12·a_1_1 − 2·b_2_213·c_10_12·a_3_5
       + b_2_213·b_2_3·c_10_12·a_1_1 + b_2_214·c_10_12·a_1_1
       − 2·b_2_25·b_2_33·c_10_122·a_3_5 − b_2_26·b_2_32·c_10_122·a_3_5
       + b_2_26·b_2_33·c_10_122·a_1_1 − 2·b_2_27·b_2_3·c_10_122·a_3_5
       − 2·b_2_27·b_2_32·c_10_122·a_1_1 + b_2_28·b_2_3·c_10_122·a_1_1
       + c_10_123·a_9_11 − 2·b_2_2·b_2_32·c_10_123·a_3_5 + b_2_2·b_2_33·c_10_123·a_1_1
       − 2·b_2_22·b_2_32·c_10_123·a_1_1 − 2·b_2_23·c_10_123·a_3_4
       − b_2_23·b_2_3·c_10_123·a_1_1 + 2·b_2_24·c_10_123·a_1_1
  12. c_40_1 → b_2_216·b_2_34 − 2·b_2_217·b_2_33 − 2·b_2_218·b_2_32 − 2·b_2_219·b_2_3
       − b_2_218·a_1_1·a_3_5 + b_2_218·a_1_0·a_3_5 + 2·b_2_211·b_2_34·c_10_12
       + b_2_212·b_2_33·c_10_12 + 2·b_2_213·b_2_32·c_10_12
       − b_2_212·b_2_3·c_10_12·a_1_1·a_3_5 + b_2_213·c_10_12·a_1_0·a_3_5
       + b_2_26·b_2_34·c_10_122 − 2·b_2_27·b_2_33·c_10_122
       − 2·b_2_28·b_2_32·c_10_122 − 2·b_2_29·b_2_3·c_10_122
       − 2·b_2_38·c_10_122·a_1_1·a_3_5 + 2·b_2_26·b_2_32·c_10_122·a_1_1·a_3_5
       + 2·b_2_27·b_2_3·c_10_122·a_1_1·a_3_5 + b_2_28·c_10_122·a_1_1·a_3_5
       − b_2_28·c_10_122·a_1_0·a_3_5 + b_2_28·c_10_122·a_1_0·a_3_4
       + 2·b_2_2·b_2_34·c_10_123 + b_2_22·b_2_33·c_10_123
       + 2·b_2_23·b_2_32·c_10_123 − b_2_33·c_10_123·a_1_1·a_3_5
       + b_2_2·b_2_32·c_10_123·a_1_1·a_3_5 + b_2_22·b_2_3·c_10_123·a_1_1·a_3_5
       + b_2_23·c_10_123·a_1_1·a_3_5 − b_2_23·c_10_123·a_1_0·a_3_4 − 2·c_10_124

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


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