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Mod-2-Cohomology of G2(3), a group of order 4245696

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• G2(3) is a group of order 4245696.
• The group order factors as 2⁶ · 3⁶ · 7 · 13.
• The group is defined by Group([(2,3)(4,6)(5,7)(8,11)(9,12)(10,14)(13,18)(15,20)(16,21)(17,23)(19,26)(22,30)(24,32)(25,33)(27,36)(28,37)(29,39)(31,42)(34,46)(35,47)(38,51)(40,53)(41,54)(43,57)(44,58)(45,60)(48,64)(49,65)(50,67)(52,70)(55,74)(56,75)(59,79)(61,81)(62,82)(63,84)(66,80)(68,89)(69,90)(71,93)(72,94)(73,96)(76,100)(77,101)(78,103)(83,109)(85,111)(86,112)(87,113)(88,115)(91,108)(92,119)(95,123)(97,125)(98,126)(99,128)(102,124)(104,133)(105,134)(106,135)(107,136)(110,138)(114,143)(117,144)(118,146)(121,148)(122,150)(127,156)(129,158)(130,159)(131,161)(132,163)(137,169)(139,170)(140,171)(141,173)(142,174)(145,178)(147,180)(149,179)(151,182)(152,183)(153,184)(154,185)(155,186)(157,188)(160,191)(162,192)(164,193)(165,194)(166,196)(167,197)(168,199)(172,204)(175,207)(176,208)(177,200)(181,214)(187,220)(189,223)(190,224)(195,229)(198,230)(201,233)(202,234)(203,236)(205,239)(209,242)(211,243)(212,244)(213,245)(215,247)(217,249)(218,231)(219,252)(221,222)(225,256)(226,257)(227,259)(232,264)(235,268)(237,269)(238,270)(241,273)(246,258)(248,280)(250,282)(251,283)(253,285)(254,286)(255,287)(261,279)(262,288)(263,291)(265,292)(266,293)(267,295)(271,299)(272,300)(274,302)(275,303)(276,305)(277,306)(278,307)(281,310)(284,314)(289,318)(290,309)(294,321)(296,308)(297,319)(298,324)(301,328)(304,331)(311,336)(312,337)(313,339)(315,341)(316,342)(317,343)(320,347)(322,349)(323,351)(325,334)(326,353)(327,330)(329,338)(335,359)(340,363)(345,365)(346,352)(348,366)(350,367)(354,370)(355,368)(356,372)(358,369)(360,362)(374,377)(375,376),(1,2,4)(3,5,8)(6,9,13)(7,10,15)(11,16,22)(12,17,24)(14,19,27)(18,25,34)(20,28,38)(21,29,40)(23,31,43)(26,35,48)(30,41,55)(32,44,59)(33,45,61)(36,49,66)(37,50,68)(39,52,71)(42,56,76)(46,62,83)(47,63,85)(51,69,91)(53,72,95)(54,73,97)(57,77,102)(58,78,104)(60,80,106)(64,86,109)(65,87,114)(67,88,116)(70,92,120)(74,98,127)(75,99,129)(79,105,115)(81,107,119)(82,108,137)(84,110,139)(89,117,145)(90,118,147)(93,121,149)(94,122,151)(96,124,153)(100,130,160)(101,131,162)(103,132,164)(111,140,172)(112,141,138)(113,142,175)(123,152,163)(125,154,159)(126,155,161)(128,157,189)(133,165,195)(134,166,148)(135,167,198)(136,168,200)(143,176,209)(144,177,210)(146,179,212)(150,181,215)(156,187,221)(158,190,225)(169,201,207)(170,202,235)(171,203,237)(173,205,236)(174,206,240)(178,211,214)(180,213,246)(182,194,228)(183,216,248)(184,217,250)(185,218,251)(186,219,191)(188,222,254)(192,226,258)(193,227,260)(196,230,262)(197,231,263)(199,232,265)(204,238,271)(208,241,274)(220,253,270)(223,255,288)(224,239,272)(229,261,264)(233,266,294)(234,267,296)(242,275,304)(243,276,256)(244,277,286)(245,278,308)(247,279,309)(249,281,311)(252,284,315)(257,289,302)(259,290,280)(268,297,323)(269,298,325)(273,301,329)(282,312,338)(283,313,340)(285,316,292)(287,317,344)(291,319,346)(293,320,348)(295,322,350)(299,326,354)(300,327,353)(303,330,356)(305,332,314)(306,333,358)(307,334,343)(310,335,360)(318,345,363)(321,339,362)(324,352,369)(328,355,371)(331,357,336)(337,361,347)(341,364,342)(349,351,368)(359,373,367)(365,374,377)(366,375,370)(372,376,378)]).
• It is non-abelian.
• It has 2-Rank 3.
• The centre of a Sylow 2-subgroup has rank 1.
• Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1)·((1  −  t  +  t2  −  t3  +  t4) · (1  −  t2  +  t4))

  ( − 1  +  t)3 · (1  +  t  +  t2) · (1  +  t  +  t2  +  t3  +  t4  +  t5  +  t6)

• The a-invariants are -∞,-∞,-∞,-3. 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, -3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 5 minimal generators of maximal degree 7:

1. b_3_0, an element of degree 3
2. c_4_0, a Duflot element of degree 4
3. b_5_0, an element of degree 5
4. b_6_0, an element of degree 6
5. b_7_0, an element of degree 7

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 2 minimal relations of maximal degree 12:

1. b_5_0² + b_3_0·b_7_0 + c_4_0·b_3_0²
2. b_5_0·b_7_0 + b_3_0⁴ + b_6_0·b_3_0²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 14 using the Hilbert-Poincaré criterion.
• However, the last relation was already found in degree 12 and the last generator in degree 7.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_0, an element of degree 4
  2. b_6_0, an element of degree 6
  3. b_7_0, an element of degree 7
• A Duflot regular sequence is given by c_4_0.
• The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 14].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

Expressing the generators as elements of H^*(SmallGroup(576,8282); GF(2))

1. b_3_0 → b_3_0 + b_2_0·b_1_0
2. c_4_0 → b_1_0·b_3_0 + b_1_0⁴ + b_2_0² + c_4_4
3. b_5_0 → b_2_0·b_3_0 + b_2_0·b_1_0³ + b_2_0²·b_1_0 + c_4_4·b_1_0
4. b_6_0 → b_3_1² + b_1_0³·b_3_0 + b_2_0·b_1_0·b_3_0 + b_2_0²·b_1_0² + c_4_4·b_1_0²
      + b_2_0·c_4_4
5. b_7_0 → b_2_0·b_1_0²·b_3_0 + c_4_4·b_3_1 + b_2_0·c_4_4·b_1_0

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

1. b_3_0 → 0, an element of degree 3
2. c_4_0 → c_1_0⁴, an element of degree 4
3. b_5_0 → 0, an element of degree 5
4. b_6_0 → 0, an element of degree 6
5. b_7_0 → 0, an element of degree 7

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

1. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2 + c_1_0·c_1_1² + c_1_0²·c_1_1, an element of degree 3
2. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. b_5_0 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2 + c_1_0·c_1_1⁴ + c_1_0⁴·c_1_1, an element of degree 5
4. b_6_0 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + 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
5. 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

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

1. b_3_0 → 0, an element of degree 3
2. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. b_5_0 → 0, an element of degree 5
4. b_6_0 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + 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
5. 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

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

1. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
2. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. b_5_0 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2, an element of degree 5
4. b_6_0 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + 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
5. 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

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

1. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
2. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. b_5_0 → c_1_1·c_1_2⁴ + c_1_1⁴·c_1_2, an element of degree 5
4. b_6_0 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + 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
5. 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

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

1. b_3_0 → 0, an element of degree 3
2. c_4_0 → c_1_2⁴ + c_1_1²·c_1_2² + c_1_1⁴ + c_1_0·c_1_1·c_1_2² + c_1_0·c_1_1²·c_1_2
      + c_1_0²·c_1_2² + c_1_0²·c_1_1·c_1_2 + c_1_0²·c_1_1² + c_1_0⁴, an element of degree 4
3. b_5_0 → 0, an element of degree 5
4. b_6_0 → c_1_1²·c_1_2⁴ + c_1_1⁴·c_1_2² + 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
5. 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

About the group

Ring generators

Ring relations

Completion information

Restriction maps