Finite Group DatabasebetaFinite Group Databasebeta| order | 13116 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C1093⋊C3) |
| unambiguous semidirect product | C1093⋊A4 |
| Fitting, socle, derived, Hall | C22×C1093 ⋊ C3 |
| Sylow | C1093 ⋊ A4 |
| Sylow | C22 ⋊ C1093⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C1093 | C3 |
| LCS | G ▷ C22×C1093 | |
| UCS | C1 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1097 |
| transitive degree | μt(G) = 4372 |
A minimal faithful permutation representation — 2 generators on 1097 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,376)(293,335,373)(296,338,370)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,368,340)(329,371,337)(331,374,334)(333,378,425)(336,381,422)(339,384,419)(342,387,416)(345,390,413)(348,393,410)(351,396,407)(354,399,404)(357,402,401)(360,405,398)(363,408,395)(366,411,392)(369,414,389)(372,417,386)(375,420,383)(377,423,380)(379,427,477)(382,430,474)(385,433,471)(388,436,468)(391,439,465)(394,442,462)(397,445,459)(400,448,456)(403,451,453)(406,454,450)(409,457,447)(412,460,444)(415,463,441)(418,466,438)(421,469,435)(424,472,432)(426,475,429)(428,479,532)(431,482,529)(434,485,526)(437,488,523)(440,491,520)(443,494,517)(446,497,514)(449,500,511)(452,503,508)(455,506,505)(458,509,502)(461,512,499)(464,515,496)(467,518,493)(470,521,490)(473,524,487)(476,527,484)(478,530,481)(480,534,590)(483,537,587)(486,540,584)(489,543,581)(492,546,578)(495,549,575)(498,552,572)(501,555,569)(504,558,566)(507,561,563)(510,564,560)(513,567,557)(516,570,554)(519,573,551)(522,576,548)(525,579,545)(528,582,542)(531,585,539)(533,588,536)(535,592,651)(538,595,648)(541,598,645)(544,601,642)(547,604,639)(550,607,636)(553,610,633)(556,613,630)(559,616,627)(562,619,624)(565,622,621)(568,625,618)(571,628,615)(574,631,612)(577,634,609)(580,637,606)(583,640,603)(586,643,600)(589,646,597)(591,649,594)(593,653,714)(596,656,711)(599,659,708)(602,662,705)(605,665,702)(608,668,699)(611,671,696)(614,674,694)(617,677,691)(620,680,688)(623,683,685)(626,686,682)(629,689,679)(632,692,676)(635,654,673)(638,697,670)(641,700,667)(644,703,664)(647,706,661)(650,709,658)(652,712,655)(657,717,774)(660,720,771)(663,723,768)(666,726,765)(669,729,762)(672,732,759)(675,735,757)(678,738,755)(681,741,752)(684,744,749)(687,747,746)(690,750,743)(693,753,740)(695,718,737)(698,758,734)(701,760,731)(704,763,728)(707,766,725)(710,769,722)(713,772,719)(715,775,716)(721,778,829)(724,781,826)(727,784,823)(730,787,820)(733,790,817)(736,793,815)(739,795,814)(742,798,812)(745,801,809)(748,804,806)(751,807,803)(754,810,800)(756,779,797)(761,816,792)(764,818,789)(767,821,786)(770,824,783)(773,827,780)(776,830,777)(782,833,878)(785,836,875)(788,839,872)(791,842,869)(794,845,867)(796,847,866)(799,849,865)(802,852,863)(805,855,860)(808,858,857)(811,861,854)(813,834,851)(819,868,844)(822,870,841)(825,873,838)(828,876,835)(831,879,832)(837,882,921)(840,885,918)(843,888,915)(846,891,913)(848,893,912)(850,895,911)(853,897,910)(856,900,908)(859,903,905)(862,906,902)(864,883,899)(871,914,890)(874,916,887)(877,919,884)(880,922,881)(886,925,958)(889,928,955)(892,931,953)(894,933,952)(896,935,951)(898,937,950)(901,939,949)(904,942,947)(907,945,944)(909,926,941)(917,954,930)(920,956,927)(923,959,924)(929,962,989)(932,965,987)(934,967,986)(936,969,985)(938,971,984)(940,973,983)(943,975,982)(946,978,980)(948,963,977)(957,988,964)(960,990,961)(966,993,1015)(968,995,1014)(970,997,1013)(972,999,1012)(974,1001,1011)(976,1003,1010)(979,1005,1009)(981,994,1007)(991,1016,992)(996,1017,1036)(998,1019,1035)(1000,1021,1034)(1002,1023,1033)(1004,1025,1032)(1006,1027,1031)(1008,1018,1030)(1020,1037,1053)(1022,1039,1052)(1024,1041,1051)(1026,1043,1050)(1028,1045,1049)(1029,1038,1048)(1040,1054,1067)(1042,1056,1066)(1044,1058,1065)(1046,1060,1064)(1047,1055,1063)(1057,1068,1078)(1059,1070,1077)(1061,1072,1076)(1062,1069,1075)(1071,1079,1086)(1073,1081,1085)(1074,1080,1084)(1082,1087,1091)(1083,1088,1090)(1089,1092,1093)(1095,1096,1097);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,535,593,654,573,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,734,792,844,890,930,964,992,1016,956,873,766,643,527,423,330,373,419,468,520,575,633,694,755,812,863,908,947,980,1007,962,885,784,662,540,430,332,377,424,473,525,580,638,698,714,774,829,878,921,958,989,1015,1036,1053,1067,1078,1086,1091,1093,1087,1070,1041,999,935,847,735,610,494,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,504,559,617,678,739,796,848,894,934,968,996,1018,978,903,804,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,718,631,515,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,731,789,841,887,927,961,988,916,821,703,582,472,374,287,327,370,416,465,517,572,630,691,752,809,860,905,944,977,925,836,723,598,482,378,426,476,528,583,641,701,761,775,651,711,771,826,875,918,955,987,1014,1035,1052,1066,1077,1085,1090,1079,1056,1021,969,893,793,671,549,439,341,255,181,119,146,175,206,240,277,317,360,406,455,507,562,620,681,742,799,850,896,936,970,998,1020,1038,1005,942,855,744,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,721,779,692,570,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,728,786,838,884,924,954,870,763,640,524,420,328,247,284,324,367,413,462,514,569,627,688,749,806,857,902,941,882,781,659,537,427,478,531,586,644,704,764,819,830,712,590,648,708,768,823,872,915,953,986,1013,1034,1051,1065,1076,1084,1068,1039,997,933,845,732,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,745,802,853,898,938,972,1000,1022,1040,1055,1027,975,900,801,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,724,782,834,753,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,725,783,835,881,914,818,700,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,746,803,854,899,833,720,595,479,533,589,647,707,767,822,871,879,772,649,532,587,645,705,765,820,869,913,952,985,1012,1033,1050,1064,1075,1054,1019,967,891,790,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,748,805,856,901,940,974,1002,1024,1042,1057,1069,1045,1003,939,852,741,616,500,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,727,785,837,883,810,689,567,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,542,600,661,722,780,832,868,760,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,743,800,851,778,656,534,591,650,710,770,825,874,917,922,827,709,588,477,529,584,642,702,762,817,867,912,951,984,1011,1032,1049,1063,1037,995,931,842,729,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,751,808,859,904,943,976,1004,1026,1044,1059,1071,1080,1060,1025,973,897,798,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,730,788,840,886,926,861,750,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,719,777,816,697,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,740,797,717,592,652,713,773,828,877,920,957,959,876,769,646,530,425,474,526,581,639,699,759,815,866,911,950,983,1010,1031,1048,1017,965,888,787,665,543,433,335,249,288,329,372,418,467,519,574,632,693,754,811,862,907,946,979,1006,1028,1046,1061,1073,1082,1088,1072,1043,1001,937,849,738,613,497,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,498,553,611,672,733,791,843,889,929,963,906,807,686,564,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,536,594,655,716,758,634,518,414,322,242,174,117,142,170,201,235,272,312,355,401,450,502,557,615,676,737,653,715,776,831,880,923,960,991,990,919,824,706,585,475,376,422,471,523,578,636,696,757,814,865,910,949,982,1009,1030,993,928,839,726,601,485,381,289,331,375,421,470,522,577,635,695,756,813,864,909,948,981,1008,1029,1047,1062,1074,1083,1089,1092,1081,1058,1023,971,895,795,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,736,794,846,892,932,966,994,945,858,747,622,506,402,310,230,162,106,62,30,10)(1094,1097)(1095,1096)Polycyclic presentation:
3,2^2,1093|b^a=c,c^a=bc,d^a=d^1511460 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13116, 13);