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Singular

Gap

Mod-2-Cohomology of G2(5), a group of order 5859000000

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• G2(5) is a group of order 5859000000.
• The group order factors as 2⁶ · 3³ · 5⁶ · 7 · 31.
• The group is defined by Group([(1,2553)(2,540)(3,1456)(4,3056)(5,323)(6,45)(7,3119)(8,2022)(9,2681)(10,332)(11,3614)(12,2337)(13,1216)(14,3852)(15,3291)(16,2055)(17,2043)(18,2632)(19,3229)(20,3036)(21,3159)(22,3456)(23,1834)(24,3764)(25,1268)(26,3457)(27,2790)(28,2068)(29,225)(30,3071)(31,2763)(32,947)(33,3020)(34,3233)(35,699)(36,1948)(37,2997)(38,1788)(39,502)(40,2768)(41,1605)(42,555)(43,2362)(44,302)(46,66)(47,2487)(48,2571)(49,3699)(50,938)(51,3043)(52,1458)(53,1699)(54,2004)(55,1274)(56,880)(57,1758)(58,3493)(59,396)(60,1831)(61,3590)(62,3358)(63,796)(64,179)(65,714)(67,1276)(68,3753)(69,638)(70,3738)(71,473)(72,2414)(73,3222)(74,3665)(75,1511)(76,2537)(77,3886)(78,3804)(79,903)(80,123)(81,2294)(82,375)(83,2580)(84,3798)(85,2640)(86,720)(87,458)(88,1354)(89,1727)(90,2372)(91,1677)(92,2435)(93,1382)(94,966)(95,1644)(96,2363)(97,1171)(98,3664)(99,500)(100,2673)(101,1741)(102,2581)(103,2820)(104,2053)(105,1783)(106,1107)(107,514)(108,1655)(109,725)(110,702)(111,1412)(112,2937)(113,1151)(114,1310)(115,347)(116,2568)(117,749)(118,3044)(119,499)(120,1692)(121,642)(122,1884)(124,2802)(125,253)(126,572)(127,2412)(128,395)(129,1477)(130,2167)(131,987)(132,1247)(133,2782)(134,1705)(135,998)(136,607)(137,532)(138,1417)(139,3353)(140,2955)(141,2842)(142,1757)(143,2239)(144,2775)(145,3378)(146,892)(147,1566)(148,2925)(149,2020)(150,3007)(151,1807)(152,2602)(153,3359)(154,2097)(155,1414)(156,2088)(157,3083)(158,2800)(159,3081)(160,3525)(161,3519)(162,1307)(163,2430)(164,1238)(165,1859)(166,2828)(167,1459)(168,3107)(169,1464)(170,3144)(171,3113)(172,3467)(173,587)(174,2319)(175,1935)(176,622)(178,784)(180,1596)(181,219)(182,2913)(183,2102)(184,465)(185,978)(186,1719)(187,405)(188,3731)(189,234)(190,1491)(191,1994)(192,2215)(193,3581)(194,1348)(195,379)(196,751)(197,3188)(198,2271)(199,3694)(200,3876)(201,773)(202,1413)(203,1518)(204,3196)(205,877)(206,1949)(207,3235)(208,1007)(209,606)(210,388)(211,2459)(212,657)(213,1353)(214,1716)(215,2509)(216,2507)(217,1393)(218,460)(220,2370)(221,430)(222,1752)(223,746)(224,1898)(226,645)(227,1118)(228,3298)(229,2172)(230,2404)(231,1137)(232,2030)(233,937)(235,1955)(236,2040)(237,581)(238,3563)(239,290)(240,1802)(241,3480)(242,1243)(243,815)(244,2212)(245,3191)(246,647)(247,1858)(248,3295)(249,1623)(250,2871)(251,3638)(252,2114)(254,2607)(255,698)(256,3770)(257,367)(258,2978)(259,3801)(260,3330)(261,964)(262,2518)(263,3134)(264,1101)(265,2021)(266,2103)(267,472)(268,530)(269,1862)(270,923)(271,1610)(272,3720)(273,3302)(274,1168)(275,905)(276,2926)(277,1026)(278,2696)(279,2671)(280,3701)(281,1099)(282,1930)(283,1766)(284,2942)(285,2067)(286,3294)(287,3464)(288,1526)(289,522)(291,2703)(292,3490)(293,3243)(294,941)(295,1133)(296,3357)(297,2909)(298,3264)(299,824)(300,2136)(301,350)(303,2133)(304,3659)(305,2556)(306,1576)(307,2152)(308,2140)(309,349)(310,1051)(311,1558)(312,1476)(313,3678)(314,2728)(315,3219)(316,1763)(317,1609)(318,330)(319,2586)(320,2023)(321,1430)(324,2041)(325,1420)(326,2056)(327,2118)(328,679)(329,900)(331,2708)(333,2511)(334,3169)(335,2985)(336,3823)(337,3097)(338,3863)(339,3642)(340,3091)(341,3341)(342,1679)(343,2360)(344,2542)(345,2186)(346,3594)(348,1593)(351,2761)(353,2539)(354,1108)(355,3463)(356,2764)(357,745)(358,3834)(359,423)(360,1361)(361,795)(362,3372)(363,788)(364,3269)(365,914)(368,1896)(369,1000)(370,1437)(371,1004)(372,3656)(373,838)(374,3012)(376,513)(377,3093)(378,2629)(380,3089)(381,1715)(382,3049)(383,769)(384,980)(385,1121)(386,928)(387,1429)(389,2575)(390,2202)(391,2654)(392,2545)(393,1103)(394,669)(398,3632)(399,2439)(400,1321)(401,2220)(402,2506)(403,818)(404,1252)(406,2854)(407,1598)(408,1759)(409,2368)(410,3630)(411,3728)(412,989)(413,1380)(414,845)(415,2400)(416,2354)(417,3135)(418,713)(419,3890)(420,1845)(421,2073)(422,949)(424,3588)(425,3466)(426,1652)(427,3332)(428,887)(429,2341)(431,3645)(432,3202)(433,662)(434,3179)(435,3627)(436,3215)(437,1218)(438,911)(439,498)(440,1405)(441,1212)(442,2745)(443,2217)(444,3371)(445,3871)(446,2950)(447,3460)(448,1761)(449,3639)(450,708)(451,3864)(452,1102)(453,907)(454,3237)(455,3597)(456,740)(457,962)(459,3314)(461,3355)(462,1272)(463,3197)(464,854)(466,3546)(467,469)(468,3106)(471,1424)(474,3062)(475,1376)(476,510)(477,3780)(478,1155)(479,2593)(480,2000)(481,3108)(482,1409)(483,1407)(484,1693)(485,656)(486,1284)(487,1083)(488,3759)(489,2408)(490,1756)(491,833)(492,1577)(493,1478)(494,2261)(495,627)(496,2420)(497,653)(501,3697)(503,3533)(504,3556)(505,2944)(506,2966)(507,1591)(508,1450)(509,3118)(511,1022)(512,1641)(515,2990)(516,2290)(517,2596)(518,2054)(519,1546)(520,603)(521,2691)(523,3843)(524,2544)(525,2670)(526,3066)(527,2313)(528,1929)(529,2943)(531,652)(533,1779)(534,2244)(535,2886)(536,3488)(537,3657)(538,2663)(539,1625)(541,3441)(542,3774)(543,3077)(544,1486)(545,716)(546,1131)(547,3075)(548,2283)(549,649)(550,1798)(551,3003)(552,681)(553,3741)(554,2411)(556,3061)(557,1411)(558,3637)(559,2917)(560,1554)(561,686)(562,1204)(563,1311)(564,3088)(565,2961)(566,1175)(567,3328)(568,1053)(569,1431)(570,2298)(571,359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• 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^*(Normalizer(G2(5),Centre(SylowSubgroup(G2(5),2))); 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