Mod-5-Cohomology of J3, a group of order 50232960

About the group Ring generators Ring relations Completion information Restriction maps

General information on the group

  • J3 is a group of order 50232960.
  • The group order factors as 27 · 35 · 5 · 17 · 19.
  • The group is defined by Group([(1,2787)(2,2486)(3,3616)(4,3977)(5,2766)(6,3954)(7,1919)(8,2905)(9,4774)(10,1830)(11,5026)(12,1616)(13,4348)(14,4786)(15,4610)(16,264)(17,446)(18,2767)(19,5058)(20,1813)(21,4395)(22,3190)(23,5027)(24,1220)(25,3195)(26,2542)(27,3740)(28,2103)(29,5666)(30,2838)(31,1152)(32,3891)(33,4962)(34,540)(35,1446)(36,1008)(37,3401)(38,3148)(39,5773)(40,5904)(41,2264)(42,2265)(43,1718)(44,3370)(45,263)(46,2562)(47,3198)(48,1890)(49,937)(50,2543)(51,2934)(52,2935)(53,1057)(54,5994)(55,1140)(56,2839)(57,3903)(58,435)(59,732)(60,605)(61,995)(62,3204)(63,894)(64,1219)(65,1447)(66,2143)(67,1256)(68,597)(69,1848)(70,3250)(71,950)(72,1529)(73,1053)(74,1339)(75,4674)(76,2697)(77,3875)(78,1777)(79,1719)(80,703)(81,1829)(82,3890)(83,1821)(84,2774)(85,4028)(86,4029)(87,4405)(88,2299)(89,3402)(90,938)(91,3682)(92,613)(93,3938)(94,4975)(95,4125)(96,1058)(97,2112)(98,4919)(99,4956)(100,2967)(101,1823)(102,2682)(103,3961)(104,5864)(105,6138)(106,3678)(107,3398)(108,1199)(109,2932)(110,2364)(111,1600)(112,1726)(113,576)(114,1451)(115,1666)(116,3132)(117,1685)(118,2257)(119,2365)(120,1940)(121,967)(122,3910)(123,2857)(124,598)(125,1007)(126,2760)(127,828)(128,1350)(129,3805)(130,2895)(131,1990)(132,1143)(133,1144)(134,640)(135,4675)(136,696)(137,5512)(138,3953)(139,872)(140,2721)(141,2636)(142,434)(143,6043)(144,5909)(145,899)(146,604)(147,2494)(148,1629)(149,911)(150,912)(151,3960)(152,2799)(153,3991)(154,5132)(155,4861)(156,955)(157,3285)(158,4482)(159,4584)(160,1512)(161,1372)(162,2566)(163,1592)(164,3637)(165,4797)(166,3483)(167,3484)(168,4724)(169,5204)(170,2185)(171,1690)(172,575)(173,3670)(174,4829)(175,3753)(176,1074)(177,1108)(178,4211)(179,2564)(180,895)(181,1394)(182,1763)(183,4411)(184,4746)(185,5979)(186,2876)(187,2877)(188,1845)(189,1889)(190,418)(191,3551)(192,3552)(193,1508)(194,3413)(195,1727)(196,1092)(197,1739)(198,2262)(199,3278)(200,4525)(201,5219)(202,5220)(203,821)(204,1535)(205,1478)(206,1269)(207,1750)(208,858)(209,968)(210,5543)(211,5540)(212,2100)(213,804)(214,363)(215,472)(216,2752)(217,4108)(218,2393)(219,2447)(220,2448)(221,2110)(222,5458)(223,4940)(224,1526)(225,2992)(226,1351)(227,4709)(228,4780)(229,293)(230,3807)(231,5605)(232,1970)(233,1804)(234,1805)(235,1550)(236,2398)(237,2370)(238,873)(239,2674)(240,3846)(241,3799)(242,3677)(243,4380)(244,2668)(245,1169)(246,1852)(247,5829)(248,417)(249,1939)(250,1365)(251,653)(252,1630)(253,2345)(254,2346)(255,3784)(256,1217)(257,1134)(258,1802)(259,5103)(260,1039)(261,4862)(262,3767)(265,3895)(266,3509)(267,3407)(268,5547)(269,2348)(270,1245)(271,893)(272,1450)(273,2567)(274,3909)(275,1290)(276,2020)(277,2171)(278,3232)(279,5227)(280,6098)(281,5606)(282,2459)(283,4725)(284,1809)(285,5330)(286,827)(287,5853)(288,5987)(289,2593)(290,2641)(291,2074)(292,2039)(294,386)(295,5624)(296,2111)(297,2263)(298,5705)(299,6103)(300,1109)(301,3867)(302,3742)(303,1324)(304,896)(305,517)(306,669)(307,751)(308,345)(309,5193)(310,5194)(311,5166)(312,1781)(313,3985)(314,3986)(315,4069)(316,908)(317,936)(318,2865)(319,5659)(320,648)(321,595)(322,1301)(323,2037)(324,4720)(325,730)(326,2493)(327,5351)(328,3702)(329,2648)(330,1532)(331,5907)(332,4682)(333,2667)(334,2342)(335,2798)(336,2178)(337,3244)(338,4526)(339,3513)(340,4696)(341,378)(342,578)(343,745)(344,2210)(346,521)(347,6114)(348,5390)(349,5128)(350,5129)(351,397)(352,1558)(353,1025)(354,4582)(355,4583)(356,3992)(357,2101)(358,1591)(359,1311)(360,3636)(361,3745)(362,2106)(364,5173)(365,2726)(366,1424)(367,5025)(368,2394)(369,2172)(370,3233)(371,3574)(372,3094)(373,1400)(374,4181)(375,4182)(376,4941)(377,1527)(379,4191)(380,3349)(381,5964)(382,5307)(383,1456)(384,700)(385,3024)(387,1073)(388,5379)(389,5984)(390,5428)(391,4518)(392,4658)(393,5147)(394,2737)(395,3877)(396,2755)(398,591)(399,1212)(400,1106)(401,1331)(402,3203)(403,4863)(404,4864)(405,4984)(406,2720)(407,4171)(408,845)(409,660)(410,2349)(411,1200)(412,1388)(413,3809)(414,3810)(415,2819)(416,4528)(419,5229)(420,5949)(421,4806)(422,534)(423,884)(424,654)(425,1142)(426,2308)(427,4110)(428,4953)(429,3319)(430,2127)(431,666)(432,2985)(433,1950)(436,2750)(437,5848)(438,2205)(441,4200)(442,4697)(443,3862)(444,5082)(445,5083)(447,3748)(448,4458)(449,3510)(450,1853)(451,913)(452,5541)(453,2460)(454,2289)(455,1194)(456,4994)(457,4995)(458,1596)(459,1597)(460,2261)(461,2968)(462,4831)(463,1052)(464,1684)(465,4046)(466,4021)(467,3031)(468,3710)(469,4281)(470,4434)(471,5228)(473,3084)(474,3085)(475,1268)(476,2579)(477,3858)(478,820)(479,1335)(480,5697)(481,4176)(482,3696)(483,2651)(484,3524)(485,1941)(486,6149)(487,3100)(488,4451)(489,1621)(490,2495)(491,5693)(492,5929)(493,1021)(494,4730)(495,2969)(496,651)(497,5371)(498,829)(499,1061)(500,4708)(501,2104)(502,2105)(503,1999)(504,3001)(505,5990)(506,1766)(507,2643)(508,4132)(509,4695)(510,3743)(511,1107)(512,1255)(513,779)(514,3331)(515,3023)(516,4218)(518,964)(519,4130)(520,2409)(522,848)(523,900)(524,5906)(525,3474)(526,2673)(527,2303)(528,2430)(529,3970)(530,2669)(531,3251)(532,4113)(533,3927)(535,4379)(536,2431)(537,1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  • It is non-abelian.
  • It has 5-Rank 1.
  • The centre of a Sylow 5-subgroup has rank 1.
  • Its Sylow 5-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 1.

Structure of the cohomology ring

This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(60,8); GF(5)).

General information

  • The cohomology ring is of dimension 1 and depth 1.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    ( − 1)·(1  −  t  +  t2)
    ( − 1  +  t) · (1  +  t2)
  • The a-invariants are -∞,-1. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
  • The filter degree type of any filter regular HSOP is [-1, -1].

About the group Ring generators Ring relations Completion information Restriction maps

Ring generators

The cohomology ring has 2 minimal generators of maximal degree 4:

  1. a_3_0, a nilpotent element of degree 3
  2. c_4_0, a Duflot element of degree 4

About the group Ring generators Ring relations Completion information Restriction maps

Ring relations

There is one "obvious" relation:
   a_3_02

Apart from that, there are no relations.

About the group Ring generators Ring relations Completion information Restriction maps

Data used for the Hilbert-Poincaré test

  • We proved completion in degree 4 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_4_0, an element of degree 4
  • The above filter regular HSOP forms a Duflot regular sequence.
  • The Raw Filter Degree Type of the filter regular HSOP is [-1, 3].

About the group Ring generators Ring relations Completion information Restriction maps

Restriction maps

Expressing the generators as elements of H*(SmallGroup(60,8); GF(5))

  1. a_3_0 → a_3_0
  2. c_4_0 → c_4_0

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

  1. a_3_0 → c_2_0·a_1_0, an element of degree 3
  2. c_4_0 → c_2_02, an element of degree 4

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