Mod-3-Cohomology of Normalizer(SymplecticGroup(8,2),Centre(SylowSubgroup(SymplecticGroup(8,2),3))), a group of order 7776

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • Normalizer(SymplecticGroup(8,2),Centre(SylowSubgroup(SymplecticGroup(8,2),3))) is a group of order 7776.
  • The group order factors as 25 · 35.
  • The group is defined by Group([(1,30,175)(2,220,199)(3,166,68)(4,196,209)(5,222,188)(6,71,248)(7,162,75)(8,208,16)(9,20,36)(10,124,72)(11,58,156)(12,27,167)(13,85,216)(14,180,155)(15,145,164)(17,76,117)(18,232,194)(19,255,249)(21,234,223)(22,143,147)(23,154,230)(24,115,148)(25,73,190)(26,33,224)(28,146,78)(29,233,158)(31,184,189)(32,245,219)(34,241,83)(35,174,133)(37,165,96)(38,80,177)(39,149,77)(40,229,79)(41,101,102)(42,119,82)(43,49,122)(44,253,54)(45,212,206)(46,168,191)(47,48,86)(50,204,93)(51,228,116)(52,193,88)(53,159,109)(55,70,60)(56,187,125)(57,236,183)(59,176,244)(61,251,94)(62,138,231)(63,142,118)(64,136,65)(66,235,108)(67,226,161)(69,129,218)(74,207,254)(81,135,195)(84,170,153)(87,250,210)(89,214,151)(90,242,171)(91,201,169)(92,98,107)(95,141,181)(97,182,152)(99,252,144)(100,139,246)(103,137,240)(104,173,213)(105,198,217)(106,179,157)(110,150,160)(111,186,163)(112,130,178)(113,225,127)(114,215,192)(120,123,132)(121,238,203)(126,247,134)(128,202,131)(140,239,211)(172,227,243)(185,237,197)(200,205,221),(1,175,30)(2,199,220)(3,68,166)(4,209,196)(5,188,222)(6,248,71)(7,75,162)(8,16,208)(9,36,20)(10,72,124)(11,156,58)(12,167,27)(13,216,85)(14,155,180)(15,164,145)(17,117,76)(18,194,232)(19,249,255)(21,223,234)(22,147,143)(23,230,154)(24,148,115)(25,190,73)(26,224,33)(28,78,146)(29,158,233)(31,189,184)(32,219,245)(34,83,241)(35,133,174)(37,96,165)(38,177,80)(39,77,149)(40,79,229)(41,102,101)(42,82,119)(43,122,49)(44,54,253)(45,206,212)(46,191,168)(47,86,48)(50,93,204)(51,116,228)(52,88,193)(53,109,159)(55,60,70)(56,125,187)(57,183,236)(59,244,176)(61,94,251)(62,231,138)(63,118,142)(64,65,136)(66,108,235)(67,161,226)(69,218,129)(74,254,207)(81,195,135)(84,153,170)(87,210,250)(89,151,214)(90,171,242)(91,169,201)(92,107,98)(95,181,141)(97,152,182)(99,144,252)(100,246,139)(103,240,137)(104,213,173)(105,217,198)(106,157,179)(110,160,150)(111,163,186)(112,178,130)(113,127,225)(114,192,215)(120,132,123)(121,203,238)(126,134,247)(128,131,202)(140,211,239)(172,243,227)(185,197,237)(200,221,205),(2,239,74)(3,12,142)(4,37,67)(5,11,124)(6,162,187)(7,56,248)(8,57,206)(9,78,181)(10,188,156)(13,82,185)(14,180,155)(15,218,101)(16,183,212)(17,76,117)(18,137,179)(19,255,249)(20,28,95)(21,65,53)(22,203,105)(23,154,230)(24,115,148)(25,168,31)(26,33,224)(27,118,166)(29,233,158)(32,245,219)(34,241,83)(35,221,169)(36,146,141)(38,176,111)(39,43,182)(40,228,243)(41,164,129)(42,237,85)(44,195,127)(45,208,236)(46,189,190)(47,48,86)(49,152,149)(50,204,93)(51,227,79)(52,131,242)(54,135,225)(55,70,60)(58,72,222)(59,163,177)(61,251,94)(62,213,170)(63,68,167)(64,159,234)(66,235,108)(69,102,145)(71,75,125)(73,191,184)(77,122,97)(80,244,186)(81,113,253)(84,231,173)(87,247,215)(88,202,90)(89,214,151)(91,174,200)(92,98,107)(96,161,209)(99,252,144)(100,139,246)(103,106,194)(104,153,138)(109,223,136)(110,150,160)(112,130,178)(114,210,126)(116,172,229)(119,197,216)(120,123,132)(121,198,143)(128,171,193)(133,205,201)(134,192,250)(140,254,199)(147,238,217)(157,232,240)(165,226,196)(207,220,211),(2,74,239)(3,142,12)(4,67,37)(5,124,11)(6,187,162)(7,248,56)(8,206,57)(9,181,78)(10,156,188)(13,185,82)(14,155,180)(15,101,218)(16,212,183)(17,117,76)(18,179,137)(19,249,255)(20,95,28)(21,53,65)(22,105,203)(23,230,154)(24,148,115)(25,31,168)(26,224,33)(27,166,118)(29,158,233)(32,219,245)(34,83,241)(35,169,221)(36,141,146)(38,111,176)(39,182,43)(40,243,228)(41,129,164)(42,85,237)(44,127,195)(45,236,208)(46,190,189)(47,86,48)(49,149,152)(50,93,204)(51,79,227)(52,242,131)(54,225,135)(55,60,70)(58,222,72)(59,177,163)(61,94,251)(62,170,213)(63,167,68)(64,234,159)(66,108,235)(69,145,102)(71,125,75)(73,184,191)(77,97,122)(80,186,244)(81,253,113)(84,173,231)(87,215,247)(88,90,202)(89,151,214)(91,200,174)(92,107,98)(96,209,161)(99,144,252)(100,246,139)(103,194,106)(104,138,153)(109,136,223)(110,160,150)(112,178,130)(114,126,210)(116,229,172)(119,216,197)(120,132,123)(121,143,198)(128,193,171)(133,201,205)(134,250,192)(140,199,254)(147,217,238)(157,240,232)(165,196,226)(207,211,220),(3,243)(4,215)(6,195)(7,225)(9,147)(12,40)(17,204)(18,64)(19,107)(20,22)(21,106)(24,123)(25,186)(26,233)(27,229)(28,203)(29,224)(31,244)(33,158)(34,235)(35,193)(36,143)(37,87)(38,46)(39,84)(43,231)(44,187)(49,62)(50,117)(51,63)(52,133)(53,103)(54,56)(55,61)(59,184)(60,94)(65,194)(66,83)(67,247)(68,227)(70,251)(71,81)(73,163)(75,113)(76,93)(77,153)(78,238)(79,167)(80,168)(88,174)(90,91)(92,255)(95,105)(96,210)(97,104)(98,249)(99,112)(100,110)(108,241)(109,240)(111,190)(114,209)(115,132)(116,118)(120,148)(121,146)(122,138)(125,253)(126,161)(127,162)(128,221)(130,252)(131,205)(134,226)(135,248)(136,232)(137,159)(139,150)(141,198)(142,228)(144,178)(149,170)(152,213)(157,223)(160,246)(165,250)(166,172)(169,171)(173,182)(176,189)(177,191)(179,234)(181,217)(192,196)(200,202)(201,242),(3,243,12,40,142,228)(4,13,143,208,176,164)(5,232,11,240,124,157)(6,195,153,77,171,169)(7,225,62,49,52,133)(8,59,145,209,216,22)(9,192,31)(10,179,188,18,156,137)(14,55,214,110,86,108)(15,196,85,147,16,244)(17,117,76)(19,107,24,123,158,33)(20,114,184)(21,53,65)(25,78,250)(26,249,98,148,120,233)(27,229,118,116,166,172)(28,210,73)(29,224,255,92,115,132)(32,204,245,93,219,50)(34,246,94)(35,162,127,138,122,193)(36,215,189)(37,82,121,236,111,129)(38,41,67,185,198,45)(39,90,91,71,81,84)(42,238,183,186,218,165)(43,88,174,75,113,231)(44,104,97,128,221,187)(46,141,247)(47,66,180,70,151,150)(48,235,155,60,89,160)(51,68,227,167,79,63)(54,213,152,131,205,56)(57,163,69,96,119,203)(58,103,72,106,222,194)(61,241,100)(64,234,159)(80,101,226,237,217,212)(83,139,251)(87,190,146)(95,126,191)(99,112,252,130,144,178)(102,161,197,105,206,177)(109,136,223)(125,253,173,182,202,200)(134,168,181)(135,170,149,242,201,248),(1,175)(3,116)(4,210)(5,72)(6,225)(7,195)(8,45)(9,147)(11,222)(12,172)(13,119)(17,204)(18,64)(19,107)(20,198)(21,232)(22,141)(24,123)(25,186)(26,233)(27,228)(28,143)(29,224)(31,244)(33,158)(34,235)(35,131)(36,203)(37,126)(38,184)(39,84)(40,166)(41,145)(43,231)(44,248)(46,59)(49,153)(50,117)(51,63)(52,169)(53,157)(54,162)(55,61)(56,127)(57,208)(58,124)(60,94)(62,77)(65,240)(66,83)(67,114)(68,227)(69,164)(70,251)(71,81)(73,176)(75,113)(76,93)(78,238)(79,167)(80,168)(82,197)(87,161)(88,174)(90,91)(92,255)(95,121)(96,215)(97,170)(98,249)(99,112)(100,110)(102,129)(103,223)(104,149)(105,146)(106,136)(108,241)(109,194)(111,191)(115,132)(118,243)(120,148)(122,213)(125,253)(128,201)(130,252)(133,171)(134,226)(135,187)(137,159)(138,152)(139,150)(140,207)(142,229)(144,178)(160,246)(163,189)(165,250)(173,182)(177,190)(179,234)(181,217)(185,216)(192,196)(193,205)(199,211)(200,202)(206,236)(209,247)(220,254)(221,242),(1,175)(3,27)(4,203)(5,53)(6,213)(7,138)(8,146)(9,183)(10,234)(11,65)(12,166)(13,73)(14,34)(15,250)(16,78)(17,32)(18,179)(20,236)(21,124)(22,37)(23,230)(24,158)(25,85)(26,224)(28,208)(29,148)(31,42)(35,225)(36,57)(38,177)(39,182)(40,229)(41,126)(44,201)(45,95)(46,197)(47,251)(48,61)(49,122)(51,227)(52,193)(54,169)(55,60)(56,153)(58,136)(59,111)(62,162)(64,156)(66,150)(67,105)(68,167)(69,215)(71,173)(72,223)(74,239)(75,231)(76,219)(77,152)(81,200)(82,184)(83,180)(84,125)(86,94)(87,145)(89,100)(90,202)(91,253)(92,120)(93,204)(96,143)(97,149)(98,132)(99,252)(101,134)(102,247)(103,240)(104,248)(106,232)(107,123)(108,160)(109,222)(110,235)(112,178)(113,174)(114,129)(115,233)(116,243)(117,245)(118,142)(119,189)(121,209)(127,133)(128,242)(131,171)(135,221)(139,151)(140,207)(141,206)(147,165)(155,241)(157,194)(159,188)(161,198)(163,176)(164,210)(168,237)(170,187)(172,228)(181,212)(185,191)(186,244)(190,216)(192,218)(195,205)(196,238)(199,220)(211,254)(214,246)(217,226)(249,255),(2,3)(4,247)(5,43)(6,128)(7,88)(8,183)(9,69)(10,49)(11,182)(12,239)(13,38)(15,20)(16,206)(17,92)(18,135)(19,115)(21,221)(22,238)(23,99)(24,249)(25,184)(26,245)(27,199)(28,218)(30,175)(31,191)(32,224)(33,219)(34,151)(35,53)(36,41)(37,215)(39,124)(40,51)(42,177)(44,103)(47,55)(48,70)(50,132)(52,71)(54,179)(56,202)(57,212)(58,122)(59,237)(60,86)(62,173)(63,220)(64,205)(65,169)(67,87)(68,211)(72,97)(73,168)(74,142)(75,131)(76,98)(77,222)(78,102)(79,243)(80,197)(81,240)(82,176)(83,214)(84,213)(85,163)(89,241)(90,248)(91,223)(93,123)(95,101)(96,250)(100,150)(105,147)(106,195)(107,117)(109,200)(110,246)(111,185)(113,157)(114,226)(118,140)(119,186)(120,204)(125,242)(126,165)(127,194)(129,141)(133,234)(134,161)(136,174)(137,225)(139,160)(144,230)(145,181)(146,164)(148,255)(149,156)(152,188)(154,252)(159,201)(162,171)(166,254)(167,207)(170,231)(187,193)(192,209)(196,210)(203,217)(216,244)(227,228)(232,253),(1,175,30)(2,4,184,147,166,206,85,36,207,226,46,105,63,236,197,181,140,96,25,121,12,16,82,28)(3,183,13,95,74,37,191,238,118,8,237,146,211,196,190,22,167,45,119,9,199,161,31,198)(5,125,21,232,91,138,54,43,72,7,159,106,205,84,127,149,156,6,136,137,35,213,81,97)(10,187,109,18,221,62,113,122,11,75,65,157,174,153,135,39,222,248,234,103,201,173,44,152)(14,47,34,23,160,94,108,252)(15,116,210,80,129,243,134,176,102,79,215,163)(17,93,107,29,123,26,32,24)(19,249)(20,239,67,73,217,27,57,42,141,220,165,189,203,68,208,216,78,254,209,168,143,142,212,185)(33,219,115,117,50,98,233,120)(38,145,227,87,59,218,229,126,186,41,228,192)(40,250,111,69,51,247,177,101,172,114,244,164)(48,83,154,150,61,235,144,155)(49,188,162,223,179,169,170,253,77,124,71,53,240,200,104,225,182,58,56,64,194,133,231,195)(52,202,171)(55,214,112,246)(60,151,178,100)(66,99,180,86,241,230,110,251)(70,89,130,139)(76,204,92,158,132,224,245,148)(88,193,242,90,128,131)]).
  • It is non-abelian.
  • It has 3-Rank 4.
  • The centre of a Sylow 3-subgroup has rank 2.
  • Its Sylow 3-subgroup has 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3 and 4, respectively.

Structure of the cohomology ring

The computation was based on 15 stability conditions for H*(81gp7xC3; GF(3)).

General information

  • The cohomology ring is of dimension 4 and depth 4.
  • The depth exceeds the Duflot bound, which is 2.
  • The Poincaré series is
    (1  −  t  +  t2) · (1  −  2·t  +  3·t2  −  3·t3  +  3·t4  −  3·t5  +  4·t6  −  3·t7  +  3·t8  −  3·t9  +  3·t10  −  2·t11  +  t12)
    ( − 1  +  t)4 · (1  +  t  +  t2) · (1  +  t2)4 · (1  −  t2  +  t4)
  • The a-invariants are -∞,-∞,-∞,-∞,-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 12 minimal generators of maximal degree 12:

  1. a_3_2, a nilpotent element of degree 3
  2. a_3_1, a nilpotent element of degree 3
  3. a_3_0, a nilpotent element of degree 3
  4. c_4_2, a Duflot element of degree 4
  5. b_4_1, an element of degree 4
  6. b_4_0, an element of degree 4
  7. a_6_0, a nilpotent element of degree 6
  8. a_7_9, a nilpotent element of degree 7
  9. a_7_1, a nilpotent element of degree 7
  10. c_8_4, a Duflot element of degree 8
  11. a_11_13, a nilpotent element of degree 11
  12. c_12_12, a Duflot element of degree 12

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There are 6 "obvious" relations:
   a_3_02, a_3_12, a_3_22, a_7_12, a_7_92, a_11_132

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

  1. a_6_0·a_3_2
  2. a_3_2·a_7_9
  3. b_4_1·a_6_0 − a_3_2·a_7_1 + b_4_0·a_3_1·a_3_2 + c_4_2·a_3_1·a_3_2
  4. b_4_1·a_7_9 − c_8_4·a_3_2 − b_4_0·c_4_2·a_3_2 − c_4_22·a_3_2
  5. a_6_02
  6. a_6_0·a_7_1 + b_4_0·a_6_0·a_3_1 + c_4_2·a_6_0·a_3_1
  7. a_6_0·a_7_9
  8. a_3_2·a_11_13 − a_6_0·c_8_4 − b_4_0·c_4_2·a_6_0 − c_4_22·a_6_0
  9. a_7_1·a_7_9 + b_4_0·a_3_1·a_7_9 + a_6_0·c_8_4 + b_4_0·c_4_2·a_6_0 + c_4_2·a_3_1·a_7_9
       + c_4_22·a_6_0
  10. c_12_12·a_3_2 − c_8_4·a_7_9 − b_4_0·c_4_2·a_7_9 − c_4_22·a_7_9
  11. b_4_1·a_11_13 − c_8_4·a_7_1 − b_4_0·c_8_4·a_3_1 − b_4_0·c_4_2·a_7_1
       − b_4_02·c_4_2·a_3_1 − c_4_2·c_8_4·a_3_1 − c_4_22·a_7_1 + b_4_0·c_4_22·a_3_1
       − c_4_23·a_3_1
  12. b_4_1·c_12_12 − c_8_42 + b_4_0·c_4_2·c_8_4 − b_4_02·c_4_22 + c_4_22·c_8_4
       + b_4_0·c_4_23 − c_4_24
  13. a_6_0·a_11_13
  14. a_7_1·a_11_13 + b_4_0·a_3_1·a_11_13 + c_4_2·a_3_1·a_11_13
  15. a_7_9·a_11_13 − a_6_0·c_12_12
  16. c_12_12·a_7_1 − c_8_4·a_11_13 + b_4_0·c_12_12·a_3_1 − b_4_0·c_4_2·a_11_13
       + c_4_2·c_12_12·a_3_1 − c_4_22·a_11_13

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 21 using the Hilbert-Poincaré criterion.
  • However, the last relation was already found in degree 19 and the last generator in degree 12.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_2, an element of degree 4
    2. c_12_12, an element of degree 12
    3. b_4_03 − b_4_1·b_4_02 − b_4_13 + b_4_1·c_8_4 − b_4_1·b_4_0·c_4_2, an element of degree 12
    4. b_4_1, an element of degree 4
  • A Duflot regular sequence is given by c_4_2, c_12_12.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, -1, 28].
  • Modifying the above filter regular HSOP, we obtained the following parameters:
    1. c_4_2, an element of degree 4
    2. c_12_12, an element of degree 12
    3. b_4_0, an element of degree 4
    4. b_4_1, an element of degree 4
  • We found that there exists some HSOP over a finite extension field, in degrees 12,4,4,4.

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(81gp7xC3; GF(3))

  1. a_3_2 → b_2_4·a_1_1
  2. a_3_1 → b_2_4·a_1_2 + b_2_2·a_1_2 − b_2_2·a_1_0 − c_2_5·a_1_2 + c_2_5·a_1_1 + c_2_5·a_1_0
  3. a_3_0 → a_3_10 − b_2_2·a_1_2 + c_2_5·a_1_2 − c_2_5·a_1_0
  4. c_4_2 → b_2_22 + b_2_4·c_2_5 + b_2_2·c_2_5 + c_2_52
  5. b_4_1 → b_2_42
  6. b_4_0 → b_4_17 − b_2_22 − b_2_4·c_2_5
  7. a_6_0 → a_1_1·a_5_26 + b_4_17·a_1_1·a_1_2 − c_2_5·a_1_1·a_3_10 − b_2_4·c_2_5·a_1_1·a_1_2
  8. a_7_9 → c_6_38·a_1_1 + c_2_5·b_4_17·a_1_1 + b_2_4·c_2_52·a_1_1 + c_2_53·a_1_1
  9. a_7_1 → b_2_4·a_5_26 + c_2_5·b_4_17·a_1_2 − c_2_5·b_4_17·a_1_1 − b_2_4·c_2_5·a_3_10
       − b_2_42·c_2_5·a_1_2 − b_2_22·c_2_5·a_1_2 + b_2_22·c_2_5·a_1_0 − b_2_4·c_2_52·a_1_2
       + b_2_4·c_2_52·a_1_1 + c_2_53·a_1_2 − c_2_53·a_1_1 − c_2_53·a_1_0
  10. c_8_4 → b_2_4·c_6_38 − b_2_23·c_2_5 − c_2_52·b_4_17 + b_2_42·c_2_52 + b_2_22·c_2_52
       + b_2_2·c_2_53 − c_2_54
  11. a_11_13 → c_6_38·a_5_26 + b_4_17·c_6_38·a_1_2 + c_2_5·b_4_17·a_5_26 + c_2_5·b_4_172·a_1_2
       + b_2_2·c_6_38·a_3_9 − b_2_22·c_6_38·a_1_2 − b_2_23·c_2_5·a_3_9 + b_2_24·c_2_5·a_1_2
       − c_2_5·c_6_38·a_3_10 − c_2_52·b_4_17·a_3_10 − b_2_4·c_2_5·c_6_38·a_1_2
       + b_2_4·c_2_52·a_5_26 + b_2_2·c_2_5·c_6_38·a_1_0 − b_2_23·c_2_52·a_1_0
       + c_2_52·c_6_38·a_1_1 + c_2_53·a_5_26 + c_2_53·b_4_17·a_1_2 + c_2_53·b_4_17·a_1_1
       − b_2_4·c_2_53·a_3_10 − b_2_42·c_2_53·a_1_2 + b_2_2·c_2_53·a_3_9
       − b_2_22·c_2_53·a_1_2 − c_2_54·a_3_10 − b_2_4·c_2_54·a_1_2 + b_2_4·c_2_54·a_1_1
       + b_2_2·c_2_54·a_1_0 + c_2_55·a_1_1
  12. c_12_12 → c_6_382 − c_2_5·b_4_17·c_6_38 + c_2_52·b_4_172 + b_2_22·c_2_5·c_6_38
       + b_2_24·c_2_52 − b_2_4·c_2_52·c_6_38 − b_2_4·c_2_53·b_4_17 − c_2_53·c_6_38
       − c_2_54·b_4_17 + b_2_42·c_2_54 + b_2_22·c_2_54 − b_2_4·c_2_55 + c_2_56

Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2

  1. a_3_2 → 0, an element of degree 3
  2. a_3_1 →  − c_2_1·a_1_0, an element of degree 3
  3. a_3_0 → c_2_1·a_1_0, an element of degree 3
  4. c_4_2 → c_2_12, an element of degree 4
  5. b_4_1 → 0, an element of degree 4
  6. b_4_0 → 0, an element of degree 4
  7. a_6_0 → 0, an element of degree 6
  8. a_7_9 → 0, an element of degree 7
  9. a_7_1 → c_2_13·a_1_0, an element of degree 7
  10. c_8_4 →  − c_2_14, an element of degree 8
  11. a_11_13 → 0, an element of degree 11
  12. c_12_12 → c_2_26 − c_2_13·c_2_23 + c_2_16, an element of degree 12

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

  1. a_3_2 → 0, an element of degree 3
  2. a_3_1 →  − c_2_5·a_1_2 + c_2_5·a_1_0 + c_2_3·a_1_2 − c_2_3·a_1_0, an element of degree 3
  3. a_3_0 →  − c_2_5·a_1_0 − c_2_3·a_1_2 + c_2_3·a_1_0, an element of degree 3
  4. c_4_2 → c_2_52 + c_2_3·c_2_5 + c_2_32, an element of degree 4
  5. b_4_1 → 0, an element of degree 4
  6. b_4_0 →  − c_2_52, an element of degree 4
  7. a_6_0 → 0, an element of degree 6
  8. a_7_9 → 0, an element of degree 7
  9. a_7_1 → c_2_3·c_2_52·a_1_2 − c_2_3·c_2_52·a_1_0 − c_2_33·a_1_2 + c_2_33·a_1_0, an element of degree 7
  10. c_8_4 →  − c_2_3·c_2_53 + c_2_32·c_2_52 + c_2_33·c_2_5 − c_2_34, an element of degree 8
  11. a_11_13 → c_2_4·c_2_54·a_1_1 + c_2_4·c_2_54·a_1_0 − c_2_42·c_2_53·a_1_2
       − c_2_43·c_2_52·a_1_1 − c_2_43·c_2_52·a_1_0 + c_2_44·c_2_5·a_1_2
       + c_2_3·c_2_54·a_1_1 + c_2_3·c_2_54·a_1_0 + c_2_3·c_2_4·c_2_53·a_1_2
       + c_2_3·c_2_43·c_2_5·a_1_2 − c_2_32·c_2_53·a_1_2 − c_2_33·c_2_52·a_1_1
       − c_2_33·c_2_52·a_1_0 + c_2_33·c_2_4·c_2_5·a_1_2 + c_2_34·c_2_5·a_1_2, an element of degree 11
  12. c_12_12 → c_2_42·c_2_54 + c_2_44·c_2_52 + c_2_46 − c_2_3·c_2_4·c_2_54
       + c_2_3·c_2_43·c_2_52 + c_2_32·c_2_54 + c_2_33·c_2_4·c_2_52 − c_2_33·c_2_43
       + c_2_34·c_2_52 + c_2_36, an element of degree 12

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

  1. a_3_2 → c_2_8·a_1_2, an element of degree 3
  2. a_3_1 → c_2_8·a_1_0 + c_2_6·a_1_2 − c_2_6·a_1_0, an element of degree 3
  3. a_3_0 →  − c_2_9·a_1_3 − c_2_9·a_1_2 − c_2_8·a_1_3 + c_2_8·a_1_1 + c_2_7·a_1_2 + c_2_6·a_1_0, an element of degree 3
  4. c_4_2 → c_2_6·c_2_8 + c_2_62, an element of degree 4
  5. b_4_1 → c_2_82, an element of degree 4
  6. b_4_0 →  − c_2_92 + c_2_8·c_2_9 − c_2_7·c_2_8 − c_2_6·c_2_8, an element of degree 4
  7. a_6_0 → c_2_92·a_1_1·a_1_2 + c_2_92·a_1_0·a_1_2 − c_2_8·c_2_9·a_1_1·a_1_2
       − c_2_8·c_2_9·a_1_0·a_1_2 + c_2_7·c_2_9·a_1_2·a_1_3 + c_2_7·c_2_8·a_1_2·a_1_3
       + c_2_7·c_2_8·a_1_1·a_1_2 + c_2_7·c_2_8·a_1_0·a_1_2 + c_2_6·c_2_9·a_1_2·a_1_3
       + c_2_6·c_2_8·a_1_2·a_1_3 + c_2_6·c_2_8·a_1_1·a_1_2 + c_2_6·c_2_8·a_1_0·a_1_2, an element of degree 6
  8. a_7_9 →  − c_2_7·c_2_92·a_1_2 + c_2_7·c_2_8·c_2_9·a_1_2 + c_2_72·c_2_8·a_1_2 + c_2_73·a_1_2
       − c_2_6·c_2_92·a_1_2 + c_2_6·c_2_8·c_2_9·a_1_2 − c_2_6·c_2_7·c_2_8·a_1_2
       + c_2_62·c_2_8·a_1_2 + c_2_63·a_1_2, an element of degree 7
  9. a_7_1 →  − c_2_8·c_2_92·a_1_1 + c_2_82·c_2_9·a_1_1 + c_2_7·c_2_8·c_2_9·a_1_3
       + c_2_7·c_2_8·c_2_9·a_1_2 + c_2_7·c_2_82·a_1_3 − c_2_7·c_2_82·a_1_1
       + c_2_72·c_2_8·a_1_2 + c_2_6·c_2_92·a_1_2 − c_2_6·c_2_92·a_1_0
       + c_2_6·c_2_8·c_2_9·a_1_3 + c_2_6·c_2_8·c_2_9·a_1_0 + c_2_6·c_2_82·a_1_3
       − c_2_6·c_2_82·a_1_1 − c_2_6·c_2_82·a_1_0 − c_2_6·c_2_7·c_2_8·a_1_0
       + c_2_62·c_2_8·a_1_2 − c_2_62·c_2_8·a_1_0 − c_2_63·a_1_2 + c_2_63·a_1_0, an element of degree 7
  10. c_8_4 →  − c_2_7·c_2_8·c_2_92 + c_2_7·c_2_82·c_2_9 + c_2_72·c_2_82 + c_2_73·c_2_8
       + c_2_62·c_2_92 − c_2_62·c_2_8·c_2_9 + c_2_62·c_2_82 + c_2_62·c_2_7·c_2_8
       − c_2_64, an element of degree 8
  11. a_11_13 → c_2_7·c_2_94·a_1_1 + c_2_7·c_2_94·a_1_0 + c_2_7·c_2_8·c_2_93·a_1_1
       + c_2_7·c_2_8·c_2_93·a_1_0 + c_2_7·c_2_82·c_2_92·a_1_1
       + c_2_7·c_2_82·c_2_92·a_1_0 − c_2_72·c_2_93·a_1_3 − c_2_72·c_2_93·a_1_2
       + c_2_72·c_2_8·c_2_92·a_1_2 + c_2_72·c_2_82·c_2_9·a_1_3 − c_2_73·c_2_92·a_1_2
       − c_2_73·c_2_92·a_1_1 − c_2_73·c_2_92·a_1_0 + c_2_73·c_2_8·c_2_9·a_1_3
       − c_2_73·c_2_8·c_2_9·a_1_2 + c_2_73·c_2_8·c_2_9·a_1_1 + c_2_73·c_2_8·c_2_9·a_1_0
       + c_2_73·c_2_82·a_1_3 − c_2_73·c_2_82·a_1_1 − c_2_73·c_2_82·a_1_0
       + c_2_74·c_2_9·a_1_3 + c_2_74·c_2_9·a_1_2 + c_2_74·c_2_8·a_1_3 + c_2_74·c_2_8·a_1_2
       − c_2_74·c_2_8·a_1_1 − c_2_74·c_2_8·a_1_0 + c_2_75·a_1_2 + c_2_6·c_2_94·a_1_1
       + c_2_6·c_2_94·a_1_0 + c_2_6·c_2_8·c_2_93·a_1_1 + c_2_6·c_2_8·c_2_93·a_1_0
       + c_2_6·c_2_82·c_2_92·a_1_1 + c_2_6·c_2_82·c_2_92·a_1_0
       + c_2_6·c_2_7·c_2_93·a_1_3 + c_2_6·c_2_7·c_2_93·a_1_2
       − c_2_6·c_2_7·c_2_8·c_2_92·a_1_2 − c_2_6·c_2_7·c_2_82·c_2_9·a_1_3
       + c_2_6·c_2_73·c_2_9·a_1_3 + c_2_6·c_2_73·c_2_9·a_1_2 + c_2_6·c_2_73·c_2_8·a_1_3
       + c_2_6·c_2_73·c_2_8·a_1_2 − c_2_6·c_2_73·c_2_8·a_1_1 − c_2_6·c_2_73·c_2_8·a_1_0
       − c_2_6·c_2_74·a_1_2 − c_2_62·c_2_93·a_1_3 − c_2_62·c_2_93·a_1_2
       + c_2_62·c_2_8·c_2_92·a_1_2 + c_2_62·c_2_82·c_2_9·a_1_3 + c_2_62·c_2_73·a_1_2
       − c_2_63·c_2_92·a_1_2 − c_2_63·c_2_92·a_1_1 − c_2_63·c_2_92·a_1_0
       + c_2_63·c_2_8·c_2_9·a_1_3 − c_2_63·c_2_8·c_2_9·a_1_2 + c_2_63·c_2_8·c_2_9·a_1_1
       + c_2_63·c_2_8·c_2_9·a_1_0 + c_2_63·c_2_82·a_1_3 − c_2_63·c_2_82·a_1_1
       − c_2_63·c_2_82·a_1_0 + c_2_63·c_2_7·c_2_9·a_1_3 + c_2_63·c_2_7·c_2_9·a_1_2
       + c_2_63·c_2_7·c_2_8·a_1_3 + c_2_63·c_2_7·c_2_8·a_1_2 − c_2_63·c_2_7·c_2_8·a_1_1
       − c_2_63·c_2_7·c_2_8·a_1_0 + c_2_63·c_2_72·a_1_2 + c_2_64·c_2_9·a_1_3
       + c_2_64·c_2_9·a_1_2 + c_2_64·c_2_8·a_1_3 + c_2_64·c_2_8·a_1_2 − c_2_64·c_2_8·a_1_1
       − c_2_64·c_2_8·a_1_0 − c_2_64·c_2_7·a_1_2 + c_2_65·a_1_2, an element of degree 11
  12. c_12_12 → c_2_72·c_2_94 + c_2_72·c_2_8·c_2_93 + c_2_72·c_2_82·c_2_92
       + c_2_73·c_2_8·c_2_92 − c_2_73·c_2_82·c_2_9 + c_2_74·c_2_92
       − c_2_74·c_2_8·c_2_9 + c_2_74·c_2_82 − c_2_75·c_2_8 + c_2_76
       − c_2_6·c_2_7·c_2_94 − c_2_6·c_2_7·c_2_8·c_2_93 − c_2_6·c_2_7·c_2_82·c_2_92
       + c_2_6·c_2_73·c_2_92 − c_2_6·c_2_73·c_2_8·c_2_9 + c_2_6·c_2_73·c_2_82
       + c_2_6·c_2_74·c_2_8 + c_2_62·c_2_94 + c_2_62·c_2_8·c_2_93
       + c_2_62·c_2_82·c_2_92 − c_2_62·c_2_73·c_2_8 + c_2_63·c_2_8·c_2_92
       − c_2_63·c_2_82·c_2_9 + c_2_63·c_2_7·c_2_92 − c_2_63·c_2_7·c_2_8·c_2_9
       + c_2_63·c_2_7·c_2_82 − c_2_63·c_2_72·c_2_8 − c_2_63·c_2_73 + c_2_64·c_2_92
       − c_2_64·c_2_8·c_2_9 + c_2_64·c_2_82 + c_2_64·c_2_7·c_2_8 − c_2_65·c_2_8 + c_2_66, an element of degree 12

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