Finite Group DatabasebetaFinite Group Databasebeta| order | 13476 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊(C1123⋊C3) |
| unambiguous semidirect product | C1123⋊A4 |
| Fitting, socle, derived, Hall | C22×C1123 ⋊ C3 |
| Sylow | C1123 ⋊ A4 |
| Sylow | C22 ⋊ C1123⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C1123 | C3 |
| LCS | G ▷ C22×C1123 | |
| UCS | C1 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1127 |
| transitive degree | μt(G) = 4492 |
A minimal faithful permutation representation — 2 generators on 1127 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)(593,653,715)(596,656,712)(599,659,709)(602,662,706)(605,665,703)(608,668,700)(611,671,697)(614,674,694)(617,677,691)(620,680,688)(623,683,685)(626,686,682)(629,689,679)(632,692,676)(635,695,673)(638,698,670)(641,701,667)(644,704,664)(647,707,661)(650,710,658)(652,713,655)(654,717,750)(657,719,778)(660,722,775)(663,725,772)(666,728,769)(669,731,766)(672,734,763)(675,737,760)(678,740,757)(681,743,754)(684,746,751)(687,749,748)(690,752,745)(693,755,742)(696,758,739)(699,761,736)(702,764,733)(705,767,730)(708,770,727)(711,773,724)(714,776,721)(716,779,718)(720,783,812)(723,785,839)(726,788,836)(729,791,833)(732,794,830)(735,797,827)(738,800,824)(741,803,821)(744,806,818)(747,809,815)(753,813,811)(756,816,808)(759,819,805)(762,822,802)(765,825,799)(768,828,796)(771,831,793)(774,834,790)(777,837,787)(780,840,784)(781,814,782)(786,844,870)(789,846,894)(792,849,891)(795,852,888)(798,855,885)(801,858,882)(804,861,879)(807,864,876)(810,867,873)(817,871,869)(820,874,866)(823,877,863)(826,880,860)(829,883,857)(832,886,854)(835,889,851)(838,892,848)(841,895,845)(842,872,843)(847,899,922)(850,901,943)(853,904,940)(856,907,937)(859,910,934)(862,913,931)(865,916,928)(868,919,925)(875,923,921)(878,926,918)(881,929,915)(884,932,912)(887,935,909)(890,938,906)(893,941,903)(896,944,900)(897,924,898)(902,948,968)(905,950,986)(908,953,983)(911,956,980)(914,959,977)(917,962,974)(920,965,971)(927,969,967)(930,972,964)(933,975,961)(936,978,958)(939,981,955)(942,984,952)(945,987,949)(946,970,947)(951,991,1008)(954,993,1023)(957,996,1020)(960,999,1017)(963,1002,1014)(966,1005,1011)(973,1009,1007)(976,1012,1004)(979,1015,1001)(982,1018,998)(985,1021,995)(988,1024,992)(989,1010,990)(994,1028,1042)(997,1030,1054)(1000,1033,1051)(1003,1036,1048)(1006,1039,1045)(1013,1043,1041)(1016,1046,1038)(1019,1049,1035)(1022,1052,1032)(1025,1055,1029)(1026,1044,1027)(1031,1059,1070)(1034,1061,1079)(1037,1064,1076)(1040,1067,1073)(1047,1071,1069)(1050,1074,1066)(1053,1077,1063)(1056,1080,1060)(1057,1072,1058)(1062,1084,1092)(1065,1086,1098)(1068,1089,1095)(1075,1093,1091)(1078,1096,1088)(1081,1099,1085)(1082,1094,1083)(1087,1103,1108)(1090,1105,1111)(1097,1109,1107)(1100,1112,1104)(1101,1110,1102)(1106,1116,1118)(1113,1119,1117)(1114,1120,1115)(1121,1123,1122)(1125,1126,1127);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,428,480,535,593,654,683,561,451,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,431,483,538,596,657,720,746,619,503,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,434,486,541,599,660,723,786,809,680,558,448,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,437,489,544,602,663,726,789,847,867,743,616,500,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,440,492,547,605,666,729,792,850,902,919,806,677,555,445,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,443,495,550,608,669,732,795,853,905,951,965,864,740,613,497,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,446,498,553,611,672,735,798,856,908,954,994,1005,916,803,674,552,442,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,449,501,556,614,675,738,801,859,911,957,997,1031,1039,962,861,737,610,494,390,298,218,150,94,118,144,172,203,237,274,314,357,403,452,504,559,617,678,741,804,862,914,960,1000,1034,1062,1067,1002,913,800,671,549,439,341,255,181,119,146,175,206,240,277,317,360,406,455,507,562,620,681,744,807,865,917,963,1003,1037,1065,1087,1089,1036,959,858,734,607,491,387,295,215,147,177,209,243,280,320,363,409,458,510,565,623,684,747,810,868,920,966,1006,1040,1068,1090,1106,1105,1064,999,910,797,668,546,436,338,252,178,211,246,283,323,366,412,461,513,568,626,687,750,812,870,922,968,1008,1042,1070,1092,1108,1118,1116,1086,1033,956,855,731,604,488,384,292,212,248,286,326,369,415,464,516,571,629,690,753,814,715,778,839,894,943,986,1023,1054,1079,1098,1111,1103,1061,996,907,794,665,543,433,335,249,288,329,372,418,467,519,574,632,693,756,817,872,779,651,712,775,836,891,940,983,1020,1051,1076,1095,1084,1030,953,852,728,601,485,381,289,331,375,421,470,522,577,635,696,759,820,875,924,840,713,590,648,709,772,833,888,937,980,1017,1048,1073,1059,993,904,791,662,540,430,332,377,424,473,525,580,638,699,762,823,878,927,970,895,776,649,532,587,645,706,769,830,885,934,977,1014,1045,1028,950,849,725,598,482,378,426,476,528,583,641,702,765,826,881,930,973,1010,944,837,710,588,477,529,584,642,703,766,827,882,931,974,1011,991,901,788,659,537,427,478,531,586,644,705,768,829,884,933,976,1013,1044,987,892,773,646,530,425,474,526,581,639,700,763,824,879,928,971,948,846,722,595,479,533,589,647,708,771,832,887,936,979,1016,1047,1072,1024,941,834,707,585,475,376,422,471,523,578,636,697,760,821,876,925,899,785,656,534,591,650,711,774,835,890,939,982,1019,1050,1075,1094,1055,984,889,770,643,527,423,330,373,419,468,520,575,633,694,757,818,873,844,719,592,652,714,777,838,893,942,985,1022,1053,1078,1097,1110,1080,1021,938,831,704,582,472,374,287,327,370,416,465,517,572,630,691,754,815,783,653,716,780,841,896,945,988,1025,1056,1081,1100,1113,1120,1099,1052,981,886,767,640,524,420,328,247,284,324,367,413,462,514,569,627,688,751,717,781,842,897,946,989,1026,1057,1082,1101,1114,1121,1123,1112,1077,1018,935,828,701,579,469,371,285,210,244,281,321,364,410,459,511,566,624,685,748,811,869,921,967,1007,1041,1069,1091,1107,1117,1122,1119,1096,1049,978,883,764,637,521,417,325,245,176,207,241,278,318,361,407,456,508,563,621,682,745,808,866,918,964,1004,1038,1066,1088,1104,1115,1109,1074,1015,932,825,698,576,466,368,282,208,145,173,204,238,275,315,358,404,453,505,560,618,679,742,805,863,915,961,1001,1035,1063,1085,1102,1093,1046,975,880,761,634,518,414,322,242,174,117,142,170,201,235,272,312,355,401,450,502,557,615,676,739,802,860,912,958,998,1032,1060,1083,1071,1012,929,822,695,573,463,365,279,205,143,92,114,139,167,198,232,269,309,352,398,447,499,554,612,673,736,799,857,909,955,995,1029,1058,1043,972,877,758,631,515,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,444,496,551,609,670,733,796,854,906,952,992,1027,1009,926,819,692,570,460,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,441,493,548,606,667,730,793,851,903,949,990,969,874,755,628,512,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,438,490,545,603,664,727,790,848,900,947,923,816,689,567,457,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,435,487,542,600,661,724,787,845,898,871,752,625,509,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,432,484,539,597,658,721,784,843,813,686,564,454,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,429,481,536,594,655,718,782,749,622,506,402,310,230,162,106,62,30,10)(1124,1127)(1125,1126)Polycyclic presentation:
3,2^2,1123|b^a=c,c^a=bc,d^a=d^331500 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13476, 10);