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Singular

Gap

Mod-2-Cohomology of Normalizer(J3.2,Centre(SylowSubgroup(J3.2,2))), a group of order 3840

About the group

Ring generators

Ring relations

Completion information

Restriction maps

General information on the group

• Normalizer(J3.2,Centre(SylowSubgroup(J3.2,2))) is a group of order 3840.
• The group order factors as 2⁸ · 3 · 5.
• The group is defined by Group([(1,3676)(2,4742)(3,3235)(4,1713)(5,5897)(6,5865)(7,643)(8,2232)(9,2846)(10,5394)(11,4953)(12,4869)(13,4271)(14,1492)(15,3794)(16,4822)(17,2703)(18,4715)(19,5706)(20,3566)(21,2667)(22,91)(23,4648)(24,2685)(25,5653)(26,1023)(27,2156)(28,3291)(29,5904)(30,3483)(31,138)(32,3975)(33,6007)(34,1430)(35,544)(36,5329)(37,2557)(38,3901)(39,5470)(40,5186)(41,5887)(42,5484)(43,1721)(44,1328)(45,206)(46,5747)(47,1220)(48,4243)(49,3701)(50,857)(51,2634)(52,4775)(53,5525)(54,5939)(55,4920)(56,5751)(57,5950)(58,918)(59,1828)(60,6140)(61,3225)(62,5485)(63,3526)(64,5927)(65,4016)(66,5413)(67,1029)(68,4839)(69,3663)(70,1639)(71,4458)(72,2856)(73,3183)(74,5796)(76,4567)(77,4413)(78,1366)(79,1136)(80,1109)(81,4924)(82,4393)(83,2009)(84,1590)(85,1739)(86,6119)(87,1847)(88,5432)(89,1646)(90,1104)(92,5909)(93,4686)(94,2402)(95,5588)(96,2201)(97,5992)(98,396)(99,864)(100,3583)(101,2868)(102,5557)(103,4800)(104,5113)(105,2863)(106,2051)(107,2130)(108,1463)(109,580)(110,2266)(111,4206)(112,1947)(113,2928)(114,236)(115,734)(116,3077)(117,2589)(118,4929)(119,3171)(120,2138)(121,199)(122,207)(123,267)(124,2465)(125,1927)(126,548)(127,2082)(128,1083)(129,4219)(131,3114)(132,3819)(133,4406)(134,3636)(135,2043)(136,3140)(137,354)(139,1810)(140,2896)(141,1908)(142,4821)(143,1376)(144,4122)(145,3743)(146,5989)(147,1781)(148,3385)(149,4745)(150,4034)(151,1162)(152,1702)(153,4296)(154,4808)(155,2390)(156,5639)(157,2788)(158,3464)(159,3420)(160,182)(161,2114)(162,3545)(163,3728)(164,3312)(165,749)(166,4790)(167,5877)(168,4250)(169,5052)(170,3446)(171,3592)(173,1924)(174,1241)(175,6049)(176,5533)(178,5554)(179,5121)(180,2002)(181,2568)(183,2563)(184,5543)(185,5739)(186,3627)(187,4492)(188,2773)(189,3830)(190,4129)(191,1817)(192,5884)(193,3903)(194,4192)(195,4563)(196,2990)(197,4504)(198,2665)(200,2368)(201,2916)(202,1502)(203,5993)(204,6135)(205,4264)(208,5369)(209,4797)(210,3957)(211,6036)(212,5717)(213,325)(214,826)(215,2646)(216,2459)(217,1113)(218,6122)(219,339)(220,4439)(221,695)(222,4304)(223,3853)(224,4734)(225,5648)(226,3389)(227,3445)(228,3890)(229,4904)(230,2536)(231,1707)(232,5561)(233,1108)(234,3522)(235,2265)(237,5046)(238,5486)(239,1932)(240,2169)(241,5081)(242,2499)(243,3742)(244,2437)(245,3363)(246,3720)(247,5787)(248,384)(249,5560)(250,1425)(251,6084)(252,3929)(253,1057)(254,5206)(255,710)(256,1421)(257,5246)(258,1822)(259,4823)(260,4155)(261,4596)(262,1998)(263,2898)(264,2642)(265,3977)(266,1048)(268,823)(269,3259)(270,3620)(271,5337)(272,3572)(273,3863)(274,5195)(275,3442)(276,1364)(277,1329)(278,426)(279,5368)(280,3724)(281,669)(282,3712)(283,1838)(284,6056)(285,460)(286,1507)(287,2810)(288,2442)(289,5703)(290,5015)(291,4316)(292,2537)(293,1517)(294,603)(295,1547)(296,2552)(297,2590)(298,2785)(299,1991)(300,5273)(301,5937)(302,5103)(303,3996)(304,5232)(305,888)(306,5143)(307,1704)(308,2985)(309,5204)(310,4557)(311,1296)(312,4185)(313,3181)(314,3924)(315,1789)(316,1567)(317,3357)(318,3748)(319,1038)(320,3027)(321,5964)(322,3317)(323,3796)(324,3632)(326,3535)(327,756)(328,616)(329,5089)(330,1437)(332,1963)(333,4386)(334,3267)(335,6021)(336,3153)(337,778)(338,2561)(340,3711)(341,482)(342,5423)(343,6052)(344,3022)(345,2301)(346,3887)(347,2840)(348,5774)(349,569)(350,873)(351,4512)(352,2621)(353,2048)(355,1586)(356,5763)(357,3729)(358,1194)(359,1776)(360,3215)(361,2503)(362,5339)(363,3493)(364,1655)(365,1119)(366,5034)(367,1916)(368,3270)(369,3818)(370,5401)(371,1799)(372,2296)(373,5968)(374,1208)(375,3081)(376,3502)(377,999)(378,5806)(379,1910)(380,1429)(381,1453)(382,4461)(383,3524)(385,5231)(386,1068)(387,4463)(388,4239)(389,2593)(390,5905)(391,3511)(392,4462)(393,4778)(394,4273)(395,4383)(397,4952)(398,4496)(399,3182)(400,1498)(401,566)(402,5702)(403,849)(404,1931)(405,3212)(406,3095)(407,976)(408,5722)(409,3868)(410,1886)(411,4973)(412,1678)(413,6124)(414,1169)(415,2314)(416,4094)(417,5324)(418,4063)(419,4726)(420,5802)(421,1018)(422,3839)(423,1053)(424,4187)(425,4382)(428,914)(429,1539)(430,4065)(431,758)(432,828)(433,4360)(434,1709)(435,5744)(436,6000)(437,2600)(438,1503)(439,5876)(440,2286)(441,2555)(442,5519)(443,5207)(444,2064)(445,1114)(446,3165)(447,4554)(448,4882)(449,4035)(450,3018)(451,472)(452,2880)(453,5689)(454,659)(455,2779)(456,4447)(457,1971)(458,4583)(459,5597)(461,2940)(462,5386)(463,4071)(464,1387)(465,1939)(466,3787)(467,5668)(468,4542)(469,4868)(470,634)(471,2263)(473,3106)(474,1094)(475,2446)(476,2476)(477,5804)(478,3859)(479,2180)(480,787)(481,5662)(483,2542)(484,2656)(485,2210)(486,3066)(487,2413)(488,4302)(489,2554)(490,5017)(491,1396)(492,3421)(493,987)(494,842)(495,3499)(496,2747)(497,4981)(498,5387)(499,5001)(500,2981)(501,5758)(502,930)(503,5810)(504,1130)(505,1664)(506,553)(507,2146)(508,1219)(509,4756)(510,3886)(511,2342)(512,5869)(513,1132)(514,935)(515,5258)(516,5341)(517,1229)(518,4173)(520,4339)(521,3808)(522,970)(523,5321)(524,3191)(525,4075)(526,5422)(527,4061)(528,2126)(529,5598)(530,1726)(531,4788)(532,4647)(533,4519)(534,5572)(535,2635)(536,4588)(537,4375)(538,1145)(539,4651)(540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3240,3326,4931,5087,2930)(800,5027,2055,4659,4997,4799,1631,5488,1105,1049,5678,6075)(815,1665,3368,4704,1546,3327,3345,2150,4535,5908,1564,1851)(816,3961,5734,4482,2032,1281,2506,3202,5781,4865,1968,3333)(817,5390,3700,3795,2431,3840,5621,3843,3408,5095,1256,5918)(832,3559,3761,1972,3041,1596,4505,1129,1359,3374,3619,5847)(850,4769,3298,4553,5418,1297,1287,1000,4198,1777,2839,3947)(853,5075,2577,4939,1798,5627,5352,4663,3827,3474,3462,1682)(858,5921,2619,4636,1346,1770,5791,4424,3454,5170,2455,5842)(859,5655,3115,2528,2137,4098,4817,3678,1043,1410,4566,5180)(860,2932,4074,4901)(866,3774,2429,5210,3812,4325,5340,4246,3565,2202,4697,3285)(870,4595,5649,3549,1089,6022,5116,5248,5854,1758,5429,1483)(872,5632,890,5782,3028,3176,2527,3534,6144,2198,3799,5635)(882,5229,5319,6046,1251,3987,5754,3850,3280,3826,5136,1622)(885,4796,2127,4349,3614,1633,1259,4770,4087,2886,1072,5746)(886,5907,2193,3087,4536,1152,5391,4324,2038,5265,1391,1940)(887,4347,4100)(899,3297,6115,1314,5416,2143,5685,2835,3321,991,3218,2659)(903,3360,4748,3946,2792,1955,4884,2505,5750,2564,3167,3058)(904,2821,2911,2753,5489,4017,3013,3992,4358,3263,1744,2120)(905,4436,4711,1952,2347,4675,3314,4306,3723,1466,6130,4556)(908,4950,1637,2594,1188,1496)(910,5857,4038,4643,2849,1280,996,3512,4698,2813,5062,3916)(912,5108,1957,1840,3180,5397,2694,1676,2229,5200,1128,1468)(913,2313,1019,5472,1432,4414,2726,1244,3275,2998,4470,1167)(915,2825,5259,5699,5457,4936,1017,6008,3682,985,5138,1278)(917,2980,934,1877,5618,3806,5638,4142,6096,6148,3820,6047)(927,4725,6132,1344,2804,5775,1193,4692,5521,2889,5327,1587)(939,3062,5839,982,1467,1884,4407,4493,1666,4750,4858,3110)(942,5997,2047,4757,1675,1698,2257,3642,3848,5919,2269,3694)(945,1795,5285,2615,1688,5076,2857,4541,2299,5307,1752,5051)(954,5184,5330,4390,4471,3813,1680,4945,3334,5099,3588,5185)(969,5039,997,2216,5328,3455,1002,3981,2179,3400,1096,5641)(990,4204,1045,1034,2362,2522,5780,1416,1051,2566,3371,4321)(993,2268,5502,2133,5886,2225,2035,3402,1769,4434,1824,3667)(994,1959,2551,4828,4985,2258,2961,5447,3364,4452,1399,1198)(1001,5141,3834,3172,4355,1674,1773,5801,3130,1508,3244,2024)(1015,2237,4024,3415,2601,1961,4729,1634,3308,1844,3160,4955)(1041,1434,1408,5906,4268,2852,2177,1442,3304,3852,1988,3931)(1056,5759,1401,3726,2115,5585,3902,3478,5823,4812,2110,4330)(1067,1317,2962,2418,4136,1921)(1076,4137,2517,4021,4855,2248,2316,4577,4958,6087,2979,1668)(1079,3382,3842,1985,4653,4066,1841,4650,3154,1480,5056,5889)(1092,1330,3985,3112,2534,5815,2319,4396,1336,2094,5247,5219)(1095,2382,5030,6023,2848,3824,3031,2412,1724,2982,5144,3453)(1099,3204,3737,2424,3538,2780,4103,5586,1918,2689,4641,3969)(1106,5504,5256,5666,1417,4585,2246,4760,3990,2752,4917,3814)(1107,2119,4546,5336,5300,5338,4842,3520,1915,3994,3641,5898)(1116,1505,2331,1706,2704,4679,1624,2742,1790,2789,3490,4498)(1120,1813,3009,2850,5430,5935,3661,2521,4384,3023,1705,1876)(1133,1600,2632,2984,3955,6079,5676,1850,3550,5719,1859,4667)(1156,2710,4371,2680,3124,5651,5508,4531,5769,5267,2820,3652)(1157,3350,2989,5515,4621,4128,3589,3695,4309,3919,1797,1652)(1175,1351,3585,1534,4861,5090,5028,4343,3036,2416,1715,3113)(1177,6050,5359,1258,3683,1748,4894,5377,3086,3927,2184,3803)(1185,5695,2004,5182,4656,6045,3219,6009,5383,2913,4586,4551)(1222,1999,4208,2001,1775,4257,6098,4341,1459,2484,2482,4623)(1224,2607,2653,4909,3705,3262,5808,3514,1488,3432,4207,1984)(1247,5493,4235,2532,2486,2364,1618,5716,2888,2956,1322,4773)(1253,4369,2614,1875,5042,3458,5518,4977,2417,6082,5374,4963)(1307,5334,2234,3543,6025,4086,3982,4178,4682,1424,4979,5254)(1308,4411,5323,3940,1783,5070,4900,5365,1891,6030,2669,5940)(1321,5040,5999,6001,2325,2245,4840,2960,4418,5562,1807,3860)(1350,5266,4489,4890,4657,3531,4627,4029,1937,3735,4255,3033)(1356,4333,4174,4815,4676,5174,1393,3668,6149,5733,3591,5119)(1395,2967,3746,2461,1658,2548,4421,4423,1885,2778,5342,3039)(1402,1569,5972,2498,3319,3307,3762,2070,3734,3472,3983,6071)(1405,3099,5475,2074,2227,2999,3060,4594,4677,4218,1556,6028)(1406,4970,3127,5985,5325,5449,4170,4787,3242,4055,3264,3892)(1407,5455,2426,2023,1914,4244,3999,1565,6107,3265,2135,1740)(1422,4426,5448,5535,2572,5053,4764,5126,1489,6067,5529,4503)(1457,2949,5236,5011)(1458,5221,4120,2895,4528,1605,4618,2617,5580,1930,2623,5367)(1473,5439,4874,5596,3451,2075,1774,2389,3080,5279,3646,4683)(1477,5110,5684,2274,3907,5601,2411,3413,3874,4771,4450,3679)(1520,2472,2475,4863)(1525,4056,4000,4670,4156,4481,2988,2576,4708,4377,5084,5162)(1543,4684,2758,4835,2466,1615,1620,2256,2933,4705,1714,2288)(1562,4278,1912,4789,3029,2997,4051,2948,3443,4318,3088,4860)(1568,5883,3190,1954,3725,4335,5436,2953,5630,4665,4844,2329)(1611,3399,4373,3988,2885,2406,1782,3396,2674,5389,4674,4454)(1616,5357,5312,1820,3149,2162,3732,3835,2525,5216,3461,4410)(1632,3349,5724,5344,5528,2207,4420,4965,4444,3953,2157,1974)(1635,2512,4456,4476,2644,4143)(1677,6153)(1700,4794,6131,3117,5209,1811,2092,4891,3866,3527,4552,5974)(1718,4101,2044,4793,3243,5410,4984,2203,4202,2309,4161,2622)(1738,4603,2457,2175,2975,5305,3283,5153,3030,2021,1977,3494)(1750,3289,2141,2761,3497,2514,3487,4672,4484,2817,4911,4940)(1772,3005,3220,3418,6151,4005,3660,4733,6095,4853,3739,6106)(1796,2794,5092,4668,3769,4905,3581,5173,4990,3752,2575,5233)(1829,5714,2637,3828,4294,3673,5316,3574,4342,2509,4527,1975)(1842,4013,2905,3094,4003,5406,1879,5962,5765,5772,5112,3232)(1882,5941,5313,2827,2526,5317,3791,5573,5033,3539,2423,2809)(2050,5870,4451,4523,2211,5845)(2063,5205,5139,4041,2433,2544,6038,4826,4946,6111,5728,2610)(2306,3457,2565,5213)(2444,4944,6072,3179,5438,5709,4941,6005,3544,4631,5672,5071)(2797,3930,4428,4664,3807,4694,3857,3479,5902,4466,5000,4197)]).
• 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, 3 and 4, respectively.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth exceeds the Duflot bound, which is 1.
• The Poincaré series is

  ( − 1)·( − 1  +  t  −  2·t2  +  t3  −  2·t4  +  t5  −  t6  +  t10)

  (1  +  t) · ( − 1  +  t)4 · (1  +  t  +  t2) · (1  +  t2)2 · (1  +  t4)

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring generators

The cohomology ring has 14 minimal generators of maximal degree 9:

1. b_1_0, an element of degree 1
2. b_2_1, an element of degree 2
3. b_3_3, an element of degree 3
4. b_3_0, an element of degree 3
5. b_4_4, an element of degree 4
6. b_4_2, an element of degree 4
7. b_5_4, an element of degree 5
8. b_5_0, an element of degree 5
9. b_6_4, an element of degree 6
10. b_7_6, an element of degree 7
11. b_7_0, an element of degree 7
12. c_8_4, a Duflot element of degree 8
13. b_8_0, an element of degree 8
14. b_9_9, an element of degree 9

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Ring relations

There are 58 minimal relations of maximal degree 18:

1. b_1_0·b_3_3
2. b_2_1·b_3_0
3. b_4_4·b_1_0
4. b_1_0·b_5_4
5. b_3_0² + b_4_2·b_1_0²
6. b_3_0·b_3_3
7. b_2_1·b_4_2·b_1_0
8. b_4_4·b_3_0
9. b_6_4·b_1_0 + b_2_1·b_5_0
10. b_1_0·b_7_6
11. b_3_0·b_5_0 + b_1_0·b_7_0 + b_1_0³·b_5_0
12. b_3_0·b_5_4
13. b_3_3·b_5_0
14. b_3_3·b_5_4 + b_4_4² + b_2_1·b_3_3² + b_2_1²·b_4_2
15. b_2_1·b_7_6 + b_2_1·b_7_0 + b_2_1·b_1_0²·b_5_0 + b_2_1·b_4_4·b_3_3 + b_2_1·b_4_2·b_3_3
16. b_4_4·b_5_0
17. b_6_4·b_3_0
18. b_6_4·b_3_3 + b_4_4·b_5_4 + b_2_1·b_7_0 + b_2_1·b_1_0²·b_5_0 + b_2_1·b_4_2·b_3_3
19. b_1_0⁴·b_5_0 + b_8_0·b_1_0
20. b_1_0·b_9_9
21. b_3_0·b_7_0 + b_1_0³·b_7_0 + b_8_0·b_1_0² + b_4_2·b_1_0·b_5_0
22. b_3_0·b_7_6
23. b_3_3·b_7_0 + b_4_4·b_6_4 + b_2_1·b_1_0³·b_5_0 + b_2_1·b_8_0 + b_2_1·b_4_2·b_4_4
       + b_2_1²·b_3_3² + b_2_1³·b_4_4 + b_2_1³·b_4_2
24. b_3_3·b_7_6 + b_4_4·b_3_3² + b_4_4·b_6_4 + b_4_2·b_3_3² + b_2_1·b_1_0³·b_5_0
       + b_2_1·b_8_0 + b_2_1·b_4_2·b_4_4 + b_2_1²·b_3_3² + b_2_1³·b_4_4 + b_2_1³·b_4_2
25. b_5_0² + b_2_1·b_1_0³·b_5_0 + b_2_1²·b_1_0·b_5_0 + c_8_4·b_1_0²
26. b_5_0·b_5_4
27. b_5_4² + b_4_4·b_6_4 + b_2_1·b_1_0³·b_5_0 + b_2_1·b_8_0 + b_2_1·b_4_4²
       + b_2_1·b_4_2·b_4_4 + b_2_1³·b_4_4 + b_2_1³·b_4_2
28. b_2_1·b_4_2·b_5_0
29. b_4_4·b_7_6 + b_4_4·b_7_0 + b_4_4²·b_3_3 + b_4_2·b_4_4·b_3_3
30. b_6_4·b_5_0 + b_2_1²·b_1_0²·b_5_0 + b_2_1³·b_5_0 + b_2_1·c_8_4·b_1_0
31. b_6_4·b_5_4 + b_4_4·b_7_0 + b_2_1·b_9_9 + b_2_1·b_3_3³ + b_2_1·b_4_4·b_5_4
       + b_2_1·b_4_2·b_5_4 + b_2_1²·b_4_2·b_3_3
32. b_8_0·b_3_3 + b_4_4·b_7_0 + b_2_1·b_3_3³ + b_2_1·b_4_2·b_5_4 + b_2_1²·b_4_4·b_3_3
33. b_1_0⁴·b_7_0 + b_8_0·b_3_0 + b_8_0·b_1_0³
34. b_6_4² + b_4_4·b_8_0 + b_2_1·b_4_4·b_3_3² + b_2_1·b_4_4·b_6_4 + b_2_1·b_4_2·b_6_4
       + b_2_1²·b_4_2² + b_2_1³·b_1_0·b_5_0 + b_2_1³·b_6_4 + b_2_1⁴·b_4_2 + b_2_1²·c_8_4
35. b_3_0·b_9_9
36. b_3_3·b_9_9 + b_3_3⁴ + b_4_4·b_8_0 + b_2_1·b_4_4·b_3_3² + b_2_1·b_4_4·b_6_4
       + b_2_1·b_4_2·b_3_3² + b_2_1·b_4_2·b_6_4 + b_2_1²·b_1_0³·b_5_0 + b_2_1²·b_8_0
       + b_2_1²·b_4_4² + b_2_1²·b_4_2² + b_2_1³·b_3_3² + b_2_1⁴·b_4_4 + b_2_1⁴·b_4_2
37. b_5_0·b_7_0 + b_2_1·b_8_0·b_1_0² + b_2_1²·b_1_0³·b_5_0 + c_8_4·b_1_0·b_3_0
       + c_8_4·b_1_0⁴
38. b_5_0·b_7_6
39. b_5_4·b_7_0 + b_4_4·b_8_0 + b_2_1·b_4_4·b_3_3² + b_2_1·b_4_4·b_6_4 + b_2_1·b_4_2·b_6_4
       + b_2_1²·b_1_0³·b_5_0 + b_2_1²·b_8_0 + b_2_1²·b_4_4² + b_2_1²·b_4_2·b_4_4
       + b_2_1²·b_4_2² + b_2_1³·b_3_3² + b_2_1⁴·b_4_4 + b_2_1⁴·b_4_2
40. b_5_4·b_7_6 + b_4_4·b_8_0 + b_4_4³ + b_4_2·b_4_4² + b_2_1·b_4_4·b_6_4
       + b_2_1·b_4_2·b_3_3² + b_2_1·b_4_2·b_6_4 + b_2_1²·b_1_0³·b_5_0 + b_2_1²·b_8_0
       + b_2_1²·b_4_4² + b_2_1³·b_3_3² + b_2_1⁴·b_4_4 + b_2_1⁴·b_4_2
41. b_6_4·b_7_0 + b_4_4·b_9_9 + b_4_4·b_3_3³ + b_2_1·b_4_4·b_7_0 + b_2_1·b_4_4²·b_3_3
       + b_2_1·b_4_2²·b_3_3 + b_2_1²·b_8_0·b_1_0 + b_2_1²·b_4_4·b_5_4 + b_2_1³·b_7_0
       + b_2_1·c_8_4·b_3_3 + b_2_1·c_8_4·b_1_0³
42. b_6_4·b_7_6 + b_4_4·b_9_9 + b_4_4·b_3_3³ + b_4_4²·b_5_4 + b_4_2·b_4_4·b_5_4
       + b_2_1·b_4_4²·b_3_3 + b_2_1·b_4_2·b_7_0 + b_2_1·b_4_2·b_4_4·b_3_3
       + b_2_1²·b_4_4·b_5_4 + b_2_1³·b_7_0 + b_2_1³·b_1_0²·b_5_0 + b_2_1·c_8_4·b_3_3
43. b_8_0·b_5_0 + b_2_1·b_8_0·b_1_0³ + b_2_1²·b_8_0·b_1_0 + c_8_4·b_1_0⁵
44. b_8_0·b_5_4 + b_4_4·b_9_9 + b_4_4·b_3_3³ + b_2_1·b_4_4²·b_3_3 + b_2_1·b_4_2·b_7_0
       + b_2_1·b_4_2·b_4_4·b_3_3 + b_2_1²·b_3_3³ + b_2_1²·b_4_4·b_5_4
45. b_6_4·b_8_0 + b_4_4²·b_3_3² + b_4_4²·b_6_4 + b_4_2·b_4_4·b_3_3²
       + b_4_2·b_4_4·b_6_4 + b_4_2²·b_3_3² + b_2_1·b_4_4³ + b_2_1·b_4_2·b_4_4²
       + b_2_1²·b_8_0·b_1_0² + b_2_1²·b_4_2·b_3_3² + b_2_1²·b_4_2·b_6_4
       + b_2_1³·b_1_0³·b_5_0 + b_2_1³·b_4_2·b_4_4 + c_8_4·b_3_3² + b_2_1·c_8_4·b_1_0⁴
       + b_2_1·b_4_4·c_8_4
46. b_5_0·b_9_9
47. b_5_4·b_9_9 + b_4_4²·b_6_4 + b_4_2·b_4_4·b_3_3² + b_4_2·b_4_4·b_6_4
       + b_4_2²·b_3_3² + b_2_1·b_3_3⁴ + b_2_1·b_4_4³ + b_2_1·b_4_2·b_8_0
       + b_2_1·b_4_2·b_4_4² + b_2_1·b_4_2²·b_4_4 + b_2_1²·b_4_4·b_3_3²
       + b_2_1²·b_4_2·b_3_3² + b_2_1³·b_4_2·b_4_4 + c_8_4·b_3_3²
48. b_7_0² + b_4_4²·b_3_3² + b_4_4²·b_6_4 + b_4_2·b_4_4·b_3_3² + b_4_2·b_4_4·b_6_4
       + b_4_2²·b_3_3² + b_2_1·b_8_0·b_1_0⁴ + b_2_1·b_4_4·b_8_0 + b_2_1·b_4_4³
       + b_2_1·b_4_2·b_8_0 + b_2_1·b_4_2²·b_4_4 + b_2_1²·b_8_0·b_1_0²
       + b_2_1²·b_4_4·b_6_4 + b_2_1²·b_4_2·b_3_3² + b_2_1³·b_1_0³·b_5_0 + b_2_1³·b_8_0
       + b_2_1³·b_4_4² + b_2_1³·b_4_2² + b_2_1⁴·b_3_3² + b_2_1⁵·b_4_4 + b_2_1⁵·b_4_2
       + c_8_4·b_3_3² + c_8_4·b_1_0⁶ + b_4_2·c_8_4·b_1_0²
49. b_7_0·b_7_6 + b_4_4²·b_3_3² + b_4_2·b_4_4·b_3_3² + b_4_2²·b_3_3²
       + b_2_1·b_4_4³ + b_2_1·b_4_2·b_4_4² + b_2_1²·b_4_4·b_3_3² + b_2_1²·b_4_4·b_6_4
       + b_2_1³·b_1_0³·b_5_0 + b_2_1³·b_8_0 + b_2_1⁴·b_3_3² + b_2_1⁵·b_4_4
       + b_2_1⁵·b_4_2 + c_8_4·b_3_3²
50. b_7_6² + b_4_4²·b_6_4 + b_4_2·b_4_4·b_3_3² + b_4_2·b_4_4·b_6_4 + b_2_1·b_4_4·b_8_0
       + b_2_1·b_4_4³ + b_2_1·b_4_2·b_8_0 + b_2_1·b_4_2²·b_4_4 + b_2_1²·b_4_4·b_6_4
       + b_2_1²·b_4_2·b_3_3² + b_2_1³·b_1_0³·b_5_0 + b_2_1³·b_8_0 + b_2_1³·b_4_4²
       + b_2_1³·b_4_2² + b_2_1⁴·b_3_3² + b_2_1⁵·b_4_4 + b_2_1⁵·b_4_2 + c_8_4·b_3_3²
51. b_6_4·b_9_9 + b_4_4²·b_7_0 + b_4_2·b_4_4·b_7_0 + b_4_2·b_4_4²·b_3_3
       + b_4_2²·b_4_4·b_3_3 + b_2_1·b_4_4·b_3_3³ + b_2_1·b_4_4²·b_5_4
       + b_2_1·b_4_2·b_3_3³ + b_2_1·b_4_2·b_4_4·b_5_4 + b_2_1·b_4_2²·b_5_4
       + b_2_1²·b_4_2·b_7_0 + b_2_1²·b_4_2²·b_3_3 + b_2_1³·b_9_9 + b_2_1³·b_3_3³
       + b_2_1³·b_4_4·b_5_4 + b_2_1³·b_4_2·b_5_4 + b_4_4·c_8_4·b_3_3 + b_2_1·c_8_4·b_5_4
52. b_8_0·b_7_0 + b_4_4²·b_7_0 + b_4_4³·b_3_3 + b_4_2·b_4_4·b_7_0 + b_4_2·b_4_4²·b_3_3
       + b_4_2²·b_4_4·b_3_3 + b_2_1·b_8_0·b_1_0⁵ + b_2_1·b_4_2·b_9_9 + b_2_1·b_4_2·b_3_3³
       + b_2_1·b_4_2·b_4_4·b_5_4 + b_2_1²·b_8_0·b_1_0³ + b_2_1²·b_4_2·b_4_4·b_3_3
       + b_2_1²·b_4_2²·b_3_3 + b_2_1³·b_4_2·b_5_4 + b_2_1⁴·b_4_2·b_3_3
       + c_8_4·b_1_0⁴·b_3_0 + c_8_4·b_1_0⁷ + b_4_4·c_8_4·b_3_3
53. b_8_0·b_7_6 + b_4_4³·b_3_3 + b_4_2·b_4_4²·b_3_3 + b_4_2²·b_4_4·b_3_3
       + b_2_1·b_4_4·b_3_3³ + b_2_1·b_4_2·b_9_9 + b_2_1·b_4_2²·b_5_4
       + b_2_1²·b_4_4²·b_3_3 + b_2_1²·b_4_2²·b_3_3 + b_2_1³·b_4_2·b_5_4
       + b_2_1⁴·b_4_2·b_3_3 + b_4_4·c_8_4·b_3_3
54. b_8_0² + b_4_4²·b_8_0 + b_4_4⁴ + b_4_2·b_4_4·b_8_0 + b_4_2·b_4_4³
       + b_4_2²·b_4_4² + b_2_1·b_8_0·b_1_0⁶ + b_2_1·b_4_4²·b_3_3²
       + b_2_1·b_4_4²·b_6_4 + b_2_1·b_4_2·b_4_4·b_3_3² + b_2_1·b_4_2²·b_6_4
       + b_2_1²·b_3_3⁴ + b_2_1²·b_8_0·b_1_0⁴ + b_2_1²·b_4_4³ + b_2_1²·b_4_2·b_4_4²
       + b_2_1²·b_4_2³ + b_2_1⁴·b_4_4² + b_2_1⁴·b_4_2² + c_8_4·b_1_0⁸ + b_4_4²·c_8_4
55. b_7_0·b_9_9 + b_4_4²·b_8_0 + b_4_2·b_4_4·b_8_0 + b_4_2·b_4_4³ + b_4_2²·b_4_4²
       + b_2_1·b_4_4²·b_3_3² + b_2_1·b_4_2²·b_3_3² + b_2_1·b_4_2²·b_6_4
       + b_2_1²·b_4_4·b_8_0 + b_2_1²·b_4_2²·b_4_4 + b_2_1³·b_4_4·b_6_4
       + b_2_1³·b_4_2·b_6_4 + b_2_1⁴·b_1_0³·b_5_0 + b_2_1⁴·b_8_0 + b_2_1⁴·b_4_4²
       + b_2_1⁴·b_4_2·b_4_4 + b_2_1⁵·b_3_3² + b_2_1⁶·b_4_4 + b_2_1⁶·b_4_2 + b_4_4²·c_8_4
       + b_2_1·c_8_4·b_3_3² + b_2_1²·b_4_2·c_8_4
56. b_7_6·b_9_9 + b_4_4·b_3_3⁴ + b_4_2·b_3_3⁴ + b_4_2·b_4_4³ + b_4_2²·b_4_4²
       + b_2_1·b_4_4²·b_6_4 + b_2_1²·b_4_4³ + b_2_1²·b_4_2·b_8_0 + b_2_1²·b_4_2·b_4_4²
       + b_2_1²·b_4_2³ + b_2_1³·b_4_4·b_3_3² + b_2_1³·b_4_4·b_6_4
       + b_2_1³·b_4_2·b_3_3² + b_2_1³·b_4_2·b_6_4 + b_2_1⁴·b_1_0³·b_5_0 + b_2_1⁴·b_8_0
       + b_2_1⁴·b_4_2·b_4_4 + b_2_1⁴·b_4_2² + b_2_1⁵·b_3_3² + b_2_1⁶·b_4_4
       + b_2_1⁶·b_4_2 + b_4_4²·c_8_4 + b_2_1·c_8_4·b_3_3² + b_2_1²·b_4_2·c_8_4
57. b_8_0·b_9_9 + b_4_4²·b_3_3³ + b_4_4³·b_5_4 + b_4_2·b_4_4·b_9_9 + b_4_2²·b_3_3³
       + b_4_2²·b_4_4·b_5_4 + b_2_1·b_3_3⁵ + b_2_1·b_4_4³·b_3_3 + b_2_1·b_4_2·b_4_4·b_7_0
       + b_2_1·b_4_2·b_4_4²·b_3_3 + b_2_1·b_4_2²·b_7_0 + b_2_1·b_4_2³·b_3_3
       + b_2_1²·b_4_4²·b_5_4 + b_2_1²·b_4_2·b_9_9 + b_2_1²·b_4_2·b_3_3³
       + b_2_1²·b_4_2·b_4_4·b_5_4 + b_2_1³·b_4_2·b_4_4·b_3_3 + b_2_1³·b_4_2²·b_3_3
       + c_8_4·b_3_3³ + b_4_4·c_8_4·b_5_4 + b_2_1·b_4_2·c_8_4·b_3_3
58. b_9_9² + b_3_3⁶ + b_4_4³·b_3_3² + b_4_2·b_4_4²·b_6_4 + b_4_2³·b_3_3²
       + b_2_1·b_4_2·b_4_4·b_8_0 + b_2_1·b_4_2·b_4_4³ + b_2_1·b_4_2²·b_4_4²
       + b_2_1²·b_4_4²·b_3_3² + b_2_1²·b_4_4²·b_6_4 + b_2_1²·b_4_2²·b_3_3²
       + b_2_1³·b_4_2·b_8_0 + b_2_1⁴·b_4_2·b_3_3² + b_2_1⁵·b_4_2·b_4_4 + b_2_1⁵·b_4_2²
       + b_4_4·c_8_4·b_3_3² + b_4_4·b_6_4·c_8_4 + b_4_2·c_8_4·b_3_3²
       + b_2_1·c_8_4·b_1_0³·b_5_0 + b_2_1·b_8_0·c_8_4 + b_2_1·b_4_4²·c_8_4
       + b_2_1·b_4_2·b_4_4·c_8_4 + b_2_1²·c_8_4·b_3_3² + b_2_1³·b_4_4·c_8_4
       + b_2_1³·b_4_2·c_8_4

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Data used for the Hilbert-Poincaré test

• We proved completion in degree 18 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_1_0³·b_5_0 + b_1_0⁸ + b_8_0 + b_4_2² + b_2_1·b_3_3² + b_2_1·b_1_0·b_5_0
        + b_2_1·b_6_4 + b_2_1²·b_4_4 + b_2_1⁴ + c_8_4, an element of degree 8
  2. b_3_3⁴ + b_4_2·b_1_0³·b_5_0 + b_4_2·b_8_0 + b_4_2²·b_1_0⁴ + b_2_1·b_8_0·b_1_0²
        + b_2_1·b_4_2·b_3_3² + b_2_1·b_4_2·b_6_4 + b_2_1²·b_8_0 + b_2_1²·b_4_2·b_4_4
        + b_2_1²·b_4_2² + b_2_1³·b_3_3² + b_2_1³·b_1_0·b_5_0 + b_2_1³·b_6_4
        + b_2_1⁴·b_1_0⁴ + b_2_1⁴·b_4_4 + c_8_4·b_1_0⁴ + b_4_2·c_8_4 + b_2_1²·c_8_4, an element of degree 12
  3. b_4_4²·b_6_4 + b_4_2²·b_3_3² + b_2_1·b_4_4·b_8_0 + b_2_1·b_4_4³
        + b_2_1·b_4_2·b_8_0 + b_2_1²·b_4_4·b_6_4 + b_2_1²·b_4_2·b_3_3²
        + b_2_1²·b_4_2·b_6_4 + b_2_1³·b_8_0 + b_2_1³·b_4_4² + b_2_1³·b_4_2²
        + b_2_1⁴·b_3_3² + b_2_1⁴·b_1_0·b_5_0 + b_2_1⁵·b_4_4 + b_2_1⁵·b_4_2 + c_8_4·b_3_3²
        + b_4_2·c_8_4·b_1_0² + b_2_1·b_4_2·c_8_4 + b_2_1²·c_8_4·b_1_0², an element of degree 14
  4. b_3_3, an element of degree 3
• A Duflot regular sequence is given by c_8_4.
• Modifying the above filter regular HSOP, we obtained the following parameters:
1. b_1_0³·b_5_0 + b_1_0⁸ + b_8_0 + b_4_2² + b_2_1·b_3_3² + b_2_1·b_1_0·b_5_0
      + b_2_1·b_6_4 + b_2_1²·b_4_4 + b_2_1⁴ + c_8_4, an element of degree 8
2. b_1_0·b_3_0 + b_1_0⁴ + b_4_4 + b_4_2 + b_2_1·b_1_0² + b_2_1², an element of degree 4
3. b_1_0² + b_2_1, an element of degree 2
4. b_3_3, an element of degree 3

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Restriction maps

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

1. b_1_0 → b_1_2
2. b_2_1 → b_1_0² + b_2_5
3. b_3_3 → b_2_5·b_1_0
4. b_3_0 → b_3_8 + b_1_1·b_1_2² + b_1_1²·b_1_2 + a_2_4·b_1_1
5. b_4_4 → b_4_12 + b_2_5·b_1_0²
6. b_4_2 → b_1_1²·b_1_2² + b_1_1⁴ + b_4_14 + a_2_4²
7. b_5_4 → b_4_12·b_1_0 + b_2_5·b_3_9 + b_2_5²·b_1_0
8. b_5_0 → b_5_20 + b_5_19 + b_1_0²·b_3_9 + b_4_14·b_1_1 + b_4_14·b_1_0
9. b_6_4 → b_1_0·b_5_19 + b_1_0³·b_3_9 + b_6_27 + b_4_14·b_1_1² + b_2_5·b_1_0⁴ + b_2_5·b_4_14
      + a_2_4·b_1_1⁴ + a_2_4·b_4_14 + a_2_4²·b_1_1²
10. b_7_6 → b_4_14·b_1_0³ + b_2_5·b_5_19 + b_2_5·b_1_0²·b_3_9
11. b_7_0 → b_7_35 + b_1_1·b_1_2·b_5_20 + b_1_0²·b_5_19 + b_4_14·b_3_9 + b_4_14·b_3_8 + b_4_12·b_3_9
       + b_2_5²·b_3_9 + b_2_5²·b_1_0³ + a_2_4·b_1_1⁵ + a_2_4²·b_1_1³
12. c_8_4 → b_1_1·b_1_2²·b_5_20 + b_1_1²·b_1_2·b_5_20 + b_4_12·b_1_0·b_3_9 + b_2_5·b_1_0·b_5_19
       + b_2_5·b_1_0⁶ + b_2_5·b_4_14·b_1_0² + b_2_5²·b_1_0·b_3_9 + b_2_5²·b_4_14
       + a_2_4·b_1_1⁶ + c_8_46
13. b_8_0 → b_3_9·b_5_19 + b_1_1·b_1_2²·b_5_20 + b_1_1²·b_1_2·b_5_20 + b_1_1³·b_5_20
       + b_1_0⁵·b_3_9 + b_8_44 + b_4_14·b_1_1⁴ + b_4_14·b_1_0·b_3_9 + b_2_5·b_1_0·b_5_19
       + b_2_5·b_1_0⁶ + b_2_5²·b_1_0·b_3_9 + b_2_5²·b_4_14 + b_2_5²·b_4_12
       + a_2_4²·b_1_1⁴
14. b_9_9 → b_6_27·b_3_9 + b_4_14·b_1_1²·b_3_8 + b_4_12·b_1_0²·b_3_9 + b_2_5·b_4_14·b_3_9
       + b_2_5²·b_5_19 + b_2_5²·b_1_0²·b_3_9 + b_2_5²·b_4_12·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_1_0 → 0, an element of degree 1
2. b_2_1 → 0, an element of degree 2
3. b_3_3 → 0, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. b_4_4 → 0, an element of degree 4
6. b_4_2 → 0, an element of degree 4
7. b_5_4 → 0, an element of degree 5
8. b_5_0 → 0, an element of degree 5
9. b_6_4 → 0, an element of degree 6
10. b_7_6 → 0, an element of degree 7
11. b_7_0 → 0, an element of degree 7
12. c_8_4 → c_1_0⁸, an element of degree 8
13. b_8_0 → 0, an element of degree 8
14. b_9_9 → 0, an element of degree 9

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

1. b_1_0 → c_1_2, an element of degree 1
2. b_2_1 → 0, an element of degree 2
3. b_3_3 → 0, an element of degree 3
4. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
5. b_4_4 → 0, an element of degree 4
6. b_4_2 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
7. b_5_4 → 0, an element of degree 5
8. b_5_0 → c_1_0·c_1_1·c_1_2³ + c_1_0·c_1_1²·c_1_2² + c_1_0²·c_1_2³
      + c_1_0²·c_1_1·c_1_2² + c_1_0²·c_1_1²·c_1_2 + c_1_0⁴·c_1_2, an element of degree 5
9. b_6_4 → 0, an element of degree 6
10. b_7_6 → 0, an element of degree 7
11. b_7_0 → c_1_0·c_1_1·c_1_2⁵ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_2⁵
       + c_1_0²·c_1_1²·c_1_2³ + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_2³
       + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7
12. c_8_4 → c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1⁴·c_1_2² + c_1_0⁴·c_1_2⁴
       + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8
13. b_8_0 → c_1_0·c_1_1·c_1_2⁶ + c_1_0·c_1_1²·c_1_2⁵ + 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_0⁴·c_1_2⁴, an element of degree 8
14. b_9_9 → 0, an element of degree 9

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

1. b_1_0 → c_1_2, an element of degree 1
2. b_2_1 → c_1_1·c_1_2 + c_1_1², an element of degree 2
3. b_3_3 → 0, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. b_4_4 → 0, an element of degree 4
6. b_4_2 → 0, an element of degree 4
7. b_5_4 → 0, an element of degree 5
8. b_5_0 → c_1_0·c_1_1·c_1_2³ + c_1_0·c_1_1²·c_1_2² + c_1_0²·c_1_2³
      + c_1_0²·c_1_1·c_1_2² + c_1_0²·c_1_1²·c_1_2 + c_1_0⁴·c_1_2, an element of degree 5
9. b_6_4 → c_1_0·c_1_1²·c_1_2³ + c_1_0·c_1_1⁴·c_1_2 + c_1_0²·c_1_1·c_1_2³
      + c_1_0²·c_1_1⁴ + c_1_0⁴·c_1_1·c_1_2 + c_1_0⁴·c_1_1², an element of degree 6
10. b_7_6 → 0, an element of degree 7
11. b_7_0 → c_1_0·c_1_1·c_1_2⁵ + c_1_0·c_1_1²·c_1_2⁴ + c_1_0²·c_1_2⁵
       + c_1_0²·c_1_1·c_1_2⁴ + c_1_0²·c_1_1²·c_1_2³ + c_1_0⁴·c_1_2³, an element of degree 7
12. c_8_4 → c_1_0·c_1_1²·c_1_2⁵ + c_1_0·c_1_1³·c_1_2⁴ + c_1_0·c_1_1⁵·c_1_2²
       + c_1_0·c_1_1⁶·c_1_2 + 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_1·c_1_2³
       + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁸, an element of degree 8
13. b_8_0 → c_1_0·c_1_1·c_1_2⁶ + c_1_0·c_1_1²·c_1_2⁵ + 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_0⁴·c_1_2⁴, an element of degree 8
14. b_9_9 → 0, an element of degree 9

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

1. b_1_0 → c_1_2, an element of degree 1
2. b_2_1 → 0, an element of degree 2
3. b_3_3 → 0, an element of degree 3
4. b_3_0 → c_1_1·c_1_2² + c_1_1²·c_1_2, an element of degree 3
5. b_4_4 → 0, an element of degree 4
6. b_4_2 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
7. b_5_4 → 0, an element of degree 5
8. b_5_0 → c_1_0·c_1_1·c_1_2³ + c_1_0·c_1_1²·c_1_2² + c_1_0²·c_1_2³
      + c_1_0²·c_1_1·c_1_2² + c_1_0²·c_1_1²·c_1_2 + c_1_0⁴·c_1_2, an element of degree 5
9. b_6_4 → 0, an element of degree 6
10. b_7_6 → 0, an element of degree 7
11. b_7_0 → c_1_0·c_1_1·c_1_2⁵ + c_1_0·c_1_1⁴·c_1_2² + c_1_0²·c_1_2⁵
       + c_1_0²·c_1_1²·c_1_2³ + c_1_0²·c_1_1⁴·c_1_2 + c_1_0⁴·c_1_2³
       + c_1_0⁴·c_1_1·c_1_2² + c_1_0⁴·c_1_1²·c_1_2, an element of degree 7
12. c_8_4 → c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1⁴·c_1_2² + c_1_0⁴·c_1_2⁴
       + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8
13. b_8_0 → c_1_0·c_1_1·c_1_2⁶ + c_1_0·c_1_1²·c_1_2⁵ + 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_0⁴·c_1_2⁴, an element of degree 8
14. b_9_9 → 0, an element of degree 9

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

1. b_1_0 → 0, an element of degree 1
2. b_2_1 → c_1_3² + c_1_2·c_1_3 + c_1_2², an element of degree 2
3. b_3_3 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
4. b_3_0 → 0, an element of degree 3
5. b_4_4 → c_1_2²·c_1_3² + c_1_2³·c_1_3 + c_1_1·c_1_3³ + c_1_1·c_1_2²·c_1_3 + c_1_1·c_1_2³
      + c_1_1²·c_1_3² + c_1_1²·c_1_2·c_1_3 + c_1_1²·c_1_2² + c_1_0·c_1_2·c_1_3²
      + c_1_0·c_1_2²·c_1_3, an element of degree 4
6. b_4_2 → 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 + c_1_1⁴, an element of degree 4
7. b_5_4 → c_1_2·c_1_3⁴ + c_1_2³·c_1_3² + c_1_1·c_1_3⁴ + c_1_1·c_1_2²·c_1_3²
      + c_1_1·c_1_2⁴ + c_1_1²·c_1_3³ + c_1_1²·c_1_2·c_1_3² + c_1_1²·c_1_2³
      + c_1_0²·c_1_2·c_1_3² + c_1_0²·c_1_2²·c_1_3, an element of degree 5
8. b_5_0 → 0, an element of degree 5
9. b_6_4 → c_1_2⁴·c_1_3² + c_1_2⁵·c_1_3 + c_1_1·c_1_3⁵ + c_1_1·c_1_2·c_1_3⁴
      + c_1_1·c_1_2²·c_1_3³ + c_1_1·c_1_2³·c_1_3² + c_1_1·c_1_2⁵ + c_1_1²·c_1_3⁴
      + c_1_1²·c_1_2·c_1_3³ + c_1_1²·c_1_2²·c_1_3² + c_1_1²·c_1_2³·c_1_3
      + c_1_1²·c_1_2⁴ + c_1_1³·c_1_2·c_1_3² + c_1_1³·c_1_2²·c_1_3
      + c_1_0·c_1_2·c_1_3⁴ + c_1_0·c_1_2³·c_1_3² + c_1_0·c_1_1·c_1_3⁴
      + c_1_0·c_1_1·c_1_2²·c_1_3² + c_1_0·c_1_1·c_1_2⁴ + c_1_0·c_1_1²·c_1_3³
      + c_1_0·c_1_1²·c_1_2·c_1_3² + c_1_0·c_1_1²·c_1_2³ + c_1_0²·c_1_3⁴
      + c_1_0²·c_1_2³·c_1_3 + c_1_0²·c_1_2⁴ + c_1_0²·c_1_1·c_1_3³
      + c_1_0²·c_1_1·c_1_2²·c_1_3 + c_1_0²·c_1_1·c_1_2³ + c_1_0²·c_1_1²·c_1_3²
      + c_1_0²·c_1_1²·c_1_2·c_1_3 + c_1_0²·c_1_1²·c_1_2² + c_1_0³·c_1_2·c_1_3²
      + c_1_0³·c_1_2²·c_1_3 + c_1_0⁴·c_1_3² + c_1_0⁴·c_1_2·c_1_3 + c_1_0⁴·c_1_2², an element of degree 6
10. b_7_6 → c_1_1·c_1_2·c_1_3⁵ + c_1_1·c_1_2³·c_1_3³ + 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_2⁴·c_1_3 + c_1_1²·c_1_2⁵
       + c_1_1⁴·c_1_3³ + c_1_1⁴·c_1_2·c_1_3² + c_1_1⁴·c_1_2³ + c_1_0·c_1_2²·c_1_3⁴
       + c_1_0·c_1_2⁴·c_1_3² + c_1_0²·c_1_2·c_1_3⁴ + c_1_0²·c_1_2⁴·c_1_3
       + c_1_0⁴·c_1_2·c_1_3² + c_1_0⁴·c_1_2²·c_1_3, an element of degree 7
11. b_7_0 → c_1_2³·c_1_3⁴ + c_1_2⁵·c_1_3² + c_1_1²·c_1_3⁵ + c_1_1²·c_1_2⁴·c_1_3
       + c_1_1²·c_1_2⁵ + c_1_1⁴·c_1_3³ + c_1_1⁴·c_1_2²·c_1_3 + c_1_1⁴·c_1_2³
       + c_1_0²·c_1_2·c_1_3⁴ + c_1_0²·c_1_2⁴·c_1_3 + c_1_0⁴·c_1_2·c_1_3²
       + c_1_0⁴·c_1_2²·c_1_3, an element of degree 7
12. c_8_4 → c_1_2⁶·c_1_3² + c_1_2⁷·c_1_3 + c_1_1·c_1_3⁷ + c_1_1·c_1_2·c_1_3⁶
       + c_1_1·c_1_2²·c_1_3⁵ + c_1_1·c_1_2⁵·c_1_3² + c_1_1·c_1_2⁷ + c_1_1²·c_1_3⁶
       + c_1_1²·c_1_2·c_1_3⁵ + c_1_1²·c_1_2²·c_1_3⁴ + c_1_1²·c_1_2³·c_1_3³
       + c_1_1²·c_1_2⁶ + c_1_1³·c_1_3⁵ + c_1_1³·c_1_2⁴·c_1_3 + c_1_1³·c_1_2⁵
       + c_1_1⁴·c_1_3⁴ + c_1_1⁴·c_1_2·c_1_3³ + c_1_1⁴·c_1_2⁴ + c_1_1⁵·c_1_3³
       + c_1_1⁵·c_1_2·c_1_3² + c_1_1⁵·c_1_2³ + c_1_1⁶·c_1_3² + c_1_1⁶·c_1_2·c_1_3
       + c_1_1⁶·c_1_2² + c_1_0·c_1_2·c_1_3⁶ + c_1_0·c_1_2²·c_1_3⁵
       + c_1_0·c_1_2³·c_1_3⁴ + c_1_0·c_1_2⁴·c_1_3³ + c_1_0·c_1_1·c_1_3⁶
       + c_1_0·c_1_1·c_1_2·c_1_3⁵ + c_1_0·c_1_1·c_1_2²·c_1_3⁴
       + c_1_0·c_1_1·c_1_2³·c_1_3³ + 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_1·c_1_2⁶ + 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_1⁴·c_1_3³ + c_1_0·c_1_1⁴·c_1_2·c_1_3²
       + c_1_0·c_1_1⁴·c_1_2³ + c_1_0²·c_1_3⁶ + c_1_0²·c_1_2·c_1_3⁵
       + c_1_0²·c_1_2²·c_1_3⁴ + c_1_0²·c_1_2³·c_1_3³ + c_1_0²·c_1_2⁴·c_1_3²
       + c_1_0²·c_1_2⁵·c_1_3 + c_1_0²·c_1_2⁶ + c_1_0⁴·c_1_2²·c_1_3²
       + c_1_0⁴·c_1_2³·c_1_3 + c_1_0⁴·c_1_1·c_1_3³ + c_1_0⁴·c_1_1·c_1_2·c_1_3²
       + c_1_0⁴·c_1_1·c_1_2³ + c_1_0⁴·c_1_1⁴ + c_1_0⁵·c_1_2·c_1_3²
       + c_1_0⁵·c_1_2²·c_1_3 + c_1_0⁸, an element of degree 8
13. b_8_0 → c_1_2⁶·c_1_3² + c_1_2⁷·c_1_3 + c_1_1·c_1_3⁷ + c_1_1·c_1_2·c_1_3⁶
       + c_1_1·c_1_2⁷ + c_1_1²·c_1_3⁶ + c_1_1²·c_1_2³·c_1_3³
       + c_1_1²·c_1_2⁴·c_1_3² + c_1_1²·c_1_2⁵·c_1_3 + c_1_1²·c_1_2⁶
       + c_1_1³·c_1_3⁵ + c_1_1³·c_1_2·c_1_3⁴ + c_1_1³·c_1_2⁵ + c_1_1⁴·c_1_3⁴
       + c_1_1⁴·c_1_2·c_1_3³ + c_1_1⁴·c_1_2⁴ + c_1_1⁵·c_1_3³ + c_1_1⁵·c_1_2·c_1_3²
       + c_1_1⁵·c_1_2³ + c_1_1⁶·c_1_3² + c_1_1⁶·c_1_2·c_1_3 + c_1_1⁶·c_1_2²
       + c_1_0·c_1_2·c_1_3⁶ + c_1_0·c_1_2²·c_1_3⁵ + c_1_0·c_1_2⁴·c_1_3³
       + c_1_0·c_1_2⁶·c_1_3 + c_1_0·c_1_1²·c_1_3⁵ + c_1_0·c_1_1²·c_1_2⁴·c_1_3
       + c_1_0·c_1_1²·c_1_2⁵ + c_1_0·c_1_1⁴·c_1_3³ + c_1_0·c_1_1⁴·c_1_2²·c_1_3
       + c_1_0·c_1_1⁴·c_1_2³ + c_1_0²·c_1_2²·c_1_3⁴ + c_1_0²·c_1_2⁵·c_1_3
       + c_1_0²·c_1_1·c_1_3⁵ + 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_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_0³·c_1_2·c_1_3⁴ + c_1_0³·c_1_2⁴·c_1_3 + c_1_0⁴·c_1_2²·c_1_3²
       + c_1_0⁴·c_1_2³·c_1_3 + c_1_0⁴·c_1_1·c_1_3³ + c_1_0⁴·c_1_1·c_1_2²·c_1_3
       + c_1_0⁴·c_1_1·c_1_2³ + 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_0⁵·c_1_2·c_1_3² + c_1_0⁵·c_1_2²·c_1_3, an element of degree 8
14. b_9_9 → c_1_1·c_1_2²·c_1_3⁶ + c_1_1·c_1_2³·c_1_3⁵ + c_1_1·c_1_2⁵·c_1_3³
       + c_1_1·c_1_2⁶·c_1_3² + c_1_1²·c_1_2·c_1_3⁶ + c_1_1²·c_1_2²·c_1_3⁵
       + c_1_1²·c_1_2⁵·c_1_3² + c_1_1²·c_1_2⁶·c_1_3 + c_1_1⁴·c_1_2²·c_1_3³
       + c_1_1⁴·c_1_2³·c_1_3² + c_1_1⁶·c_1_2·c_1_3² + c_1_1⁶·c_1_2²·c_1_3
       + c_1_0·c_1_1·c_1_2·c_1_3⁶ + c_1_0·c_1_1·c_1_2²·c_1_3⁵
       + c_1_0·c_1_1·c_1_2³·c_1_3⁴ + c_1_0·c_1_1·c_1_2⁴·c_1_3³
       + c_1_0·c_1_1·c_1_2⁵·c_1_3² + c_1_0·c_1_1·c_1_2⁶·c_1_3 + c_1_0·c_1_1²·c_1_3⁶
       + c_1_0·c_1_1²·c_1_2·c_1_3⁵ + c_1_0·c_1_1²·c_1_2³·c_1_3³
       + c_1_0·c_1_1²·c_1_2⁵·c_1_3 + c_1_0·c_1_1²·c_1_2⁶ + c_1_0·c_1_1⁴·c_1_3⁴
       + c_1_0·c_1_1⁴·c_1_2²·c_1_3² + c_1_0·c_1_1⁴·c_1_2⁴ + c_1_0²·c_1_2·c_1_3⁶
       + c_1_0²·c_1_2²·c_1_3⁵ + c_1_0²·c_1_2³·c_1_3⁴ + c_1_0²·c_1_2⁴·c_1_3³
       + c_1_0²·c_1_1·c_1_3⁶ + c_1_0²·c_1_1·c_1_2·c_1_3⁵
       + c_1_0²·c_1_1·c_1_2³·c_1_3³ + c_1_0²·c_1_1·c_1_2⁵·c_1_3
       + c_1_0²·c_1_1·c_1_2⁶ + c_1_0²·c_1_1⁴·c_1_3³ + c_1_0²·c_1_1⁴·c_1_2²·c_1_3
       + c_1_0²·c_1_1⁴·c_1_2³ + c_1_0⁴·c_1_2³·c_1_3² + c_1_0⁴·c_1_2⁴·c_1_3
       + c_1_0⁴·c_1_1·c_1_3⁴ + c_1_0⁴·c_1_1·c_1_2²·c_1_3² + c_1_0⁴·c_1_1·c_1_2⁴
       + c_1_0⁴·c_1_1²·c_1_3³ + c_1_0⁴·c_1_1²·c_1_2·c_1_3²
       + c_1_0⁴·c_1_1²·c_1_2³ + c_1_0⁶·c_1_2·c_1_3² + c_1_0⁶·c_1_2²·c_1_3, an element of degree 9

About the group

Ring generators

Ring relations

Completion information

Restriction maps