Finite Group DatabasebetaFinite Group Databasebeta| order | 13308 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#17984585 |
| Fitting, socle, derived, Hall | C3×C1109 ⋊ C4 |
| Sylow | C1109 ⋊ Dic3 |
| Sylow | C3 ⋊ #17984585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C1109 | C4 |
| prime radical, cosocle | C3×D1109 | C2 |
| LCS | G ▷ C3×C1109 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1112 |
| transitive degree | μt(G) = 3327 |
A minimal faithful permutation representation — 2 generators on 1112 points:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,91)(53,72,73,87)(57,76,77,83)(61,80,81,79)(65,84,85,75)(67,88,89,71)(70,93,94,120)(74,97,98,116)(78,101,102,112)(82,105,106,108)(86,109,110,104)(90,113,114,100)(92,117,118,96)(95,122,123,153)(99,126,127,149)(103,130,131,145)(107,134,135,141)(111,138,139,137)(115,142,143,133)(119,146,147,129)(121,150,151,125)(124,155,156,190)(128,159,160,186)(132,163,164,182)(136,167,168,178)(140,171,172,174)(144,175,176,170)(148,179,180,166)(152,183,184,162)(154,187,188,158)(157,192,193,231)(161,196,197,227)(165,200,201,223)(169,204,205,219)(173,208,209,215)(177,212,213,211)(181,216,217,207)(185,220,221,203)(189,224,225,199)(191,228,229,195)(194,233,234,276)(198,237,238,272)(202,241,242,268)(206,245,246,264)(210,249,250,260)(214,253,254,256)(218,257,258,252)(222,261,262,248)(226,265,266,244)(230,269,270,240)(232,273,274,236)(235,278,279,325)(239,282,283,321)(243,286,287,317)(247,290,291,313)(251,294,295,309)(255,298,299,305)(259,302,303,301)(263,306,307,297)(267,310,311,293)(271,314,315,289)(275,318,319,285)(277,322,323,281)(280,327,328,378)(284,331,332,374)(288,335,336,370)(292,339,340,366)(296,343,344,362)(300,347,348,358)(304,351,352,354)(308,355,356,350)(312,359,360,346)(316,363,364,342)(320,367,368,338)(324,371,372,334)(326,375,376,330)(329,380,381,435)(333,384,385,431)(337,388,389,427)(341,392,393,423)(345,396,397,419)(349,400,401,415)(353,404,405,411)(357,408,409,407)(361,412,413,403)(365,416,417,399)(369,420,421,395)(373,424,425,391)(377,428,429,387)(379,432,433,383)(382,437,438,496)(386,441,442,492)(390,445,446,488)(394,449,450,484)(398,453,454,480)(402,457,458,476)(406,461,462,472)(410,465,466,468)(414,469,470,464)(418,473,474,460)(422,477,478,456)(426,481,482,452)(430,485,486,448)(434,489,490,444)(436,493,494,440)(439,498,499,561)(443,502,503,557)(447,506,507,553)(451,510,511,549)(455,514,515,545)(459,518,519,541)(463,522,523,537)(467,526,527,533)(471,530,531,529)(475,534,535,525)(479,538,539,521)(483,542,543,517)(487,546,547,513)(491,550,551,509)(495,554,555,505)(497,558,559,501)(500,563,564,630)(504,567,568,626)(508,571,572,622)(512,575,576,618)(516,579,580,614)(520,583,584,610)(524,587,588,606)(528,591,592,602)(532,595,596,598)(536,599,600,594)(540,603,604,590)(544,607,608,586)(548,611,612,582)(552,615,616,578)(556,619,620,574)(560,623,624,570)(562,627,628,566)(565,632,633,703)(569,636,637,699)(573,640,641,695)(577,644,645,691)(581,648,649,687)(585,652,653,683)(589,656,657,679)(593,660,661,675)(597,664,665,671)(601,668,669,667)(605,672,673,663)(609,676,677,659)(613,680,681,655)(617,684,685,651)(621,688,689,647)(625,692,693,643)(629,696,697,639)(631,700,701,635)(634,705,706,780)(638,709,710,776)(642,713,714,772)(646,717,718,768)(650,721,722,764)(654,725,726,760)(658,729,730,756)(662,733,734,752)(666,737,738,748)(670,741,742,744)(674,745,746,740)(678,749,750,736)(682,753,754,732)(686,757,758,728)(690,761,762,724)(694,765,766,720)(698,769,770,716)(702,773,774,712)(704,777,778,708)(707,782,783,861)(711,786,787,857)(715,790,791,853)(719,794,795,849)(723,798,799,845)(727,802,803,841)(731,806,807,837)(735,810,811,833)(739,814,815,829)(743,818,819,825)(747,822,823,821)(751,826,827,817)(755,830,831,813)(759,834,835,809)(763,838,839,805)(767,842,843,801)(771,846,847,797)(775,850,851,793)(779,854,855,789)(781,858,859,785)(784,863,864,946)(788,867,868,942)(792,871,872,938)(796,875,876,934)(800,879,880,930)(804,883,884,926)(808,887,888,922)(812,891,892,918)(816,895,896,914)(820,899,900,910)(824,903,904,906)(828,907,908,902)(832,911,912,898)(836,915,916,894)(840,919,920,890)(844,923,924,886)(848,927,928,882)(852,931,932,878)(856,935,936,874)(860,939,940,870)(862,943,944,866)(865,948,949,1035)(869,952,953,1031)(873,956,957,1027)(877,960,961,1023)(881,964,965,1019)(885,968,969,1015)(889,972,973,1011)(893,976,977,1007)(897,980,981,1003)(901,984,985,999)(905,988,989,995)(909,992,993,991)(913,996,997,987)(917,1000,1001,983)(921,1004,1005,979)(925,1008,1009,975)(929,1012,1013,971)(933,1016,1017,967)(937,1020,1021,963)(941,1024,1025,959)(945,1028,1029,955)(947,1032,1033,951)(950,1037,1038,1046)(954,1039,1041,1105)(958,1044,1045,1103)(962,1048,1049,1101)(966,1052,1053,1099)(970,1056,1057,1097)(974,1060,1061,1095)(978,1064,1065,1093)(982,1068,1069,1091)(986,1072,1073,1089)(990,1076,1077,1087)(994,1080,1081,1083)(998,1084,1085,1079)(1002,1088,1086,1075)(1006,1090,1082,1071)(1010,1092,1078,1067)(1014,1094,1074,1063)(1018,1096,1070,1059)(1022,1098,1066,1055)(1026,1100,1062,1051)(1030,1102,1058,1047)(1034,1104,1054,1043)(1036,1106,1050,1040)(1042,1107,1108,1109)(1111,1112);(1,2,5,10,19,32,49,70,95,124,157,194,235,280,329,382,439,500,565,634,707,784,865,950,1039,948,1036,1049,961,876,795,718,645,576,511,450,393,340,291,246,205,168,135,106,81,60,43,28,41,58,79,104,133,166,203,244,289,338,391,448,509,574,643,716,793,874,959,1047,1052,960,871,786,705,781,860,941,1026,1061,973,888,807,730,657,588,523,462,405,352,303,258,217,180,147,118,91,116,145,178,215,256,301,350,403,460,521,586,655,728,805,886,971,1059,1064,972,883,798,717,640,567,498,562,629,698,771,848,929,1014,1073,985,900,819,742,669,600,535,474,417,364,315,270,229,190,227,268,313,362,415,472,533,598,667,740,817,898,983,1071,1076,984,895,810,729,652,579,510,445,384,327,379,434,491,552,617,686,759,836,917,1002,1085,997,912,831,754,681,612,547,486,429,376,325,374,427,484,545,610,679,752,829,910,995,1083,1088,996,907,822,741,664,591,522,457,396,339,286,237,192,232,275,320,369,422,479,540,605,674,747,824,905,990,1078,1009,924,843,766,693,624,559,496,557,622,691,764,841,922,1007,1091,1094,1008,919,834,753,676,603,534,469,408,351,298,249,204,163,126,93,121,152,185,222,263,308,357,410,467,528,593,662,735,812,893,978,1066,1021,936,855,778,703,776,853,934,1019,1097,1100,1020,931,846,765,688,615,546,481,420,363,310,261,216,175,138,105,76,51,30,46,65,86,111,140,173,210,251,296,345,398,455,516,581,650,723,800,881,966,1054,1033,946,1031,1103,1106,1032,943,858,777,700,627,558,493,432,375,322,273,228,187,150,117,88,63,42,25,12,3,7,14,23,36,53,74,99,128,161,198,239,284,333,386,443,504,569,638,711,788,869,954,1042,1107,1037,1045,957,872,791,714,641,572,507,446,389,336,287,242,201,164,131,102,77,56,39,26,15,24,37,54,75,100,129,162,199,240,285,334,387,444,505,570,639,712,789,870,955,1043,1048,956,867,782,862,945,1030,1057,969,884,803,726,653,584,519,458,401,348,299,254,213,176,143,114,89,66,87,112,141,174,211,252,297,346,399,456,517,582,651,724,801,882,967,1055,1060,968,879,794,713,636,563,631,702,775,852,933,1018,1069,981,896,815,738,665,596,531,470,413,360,311,266,225,188,153,186,223,264,309,358,411,468,529,594,663,736,813,894,979,1067,1072,980,891,806,725,648,575,506,441,380,436,495,556,621,690,763,840,921,1006,1081,993,908,827,750,677,608,543,482,425,372,323,276,321,370,423,480,541,606,675,748,825,906,991,1079,1084,992,903,818,737,660,587,518,453,392,335,282,233,277,324,373,426,483,544,609,678,751,828,909,994,1082,1005,920,839,762,689,620,555,494,435,492,553,618,687,760,837,918,1003,1089,1092,1004,915,830,749,672,599,530,465,404,347,294,245,200,159,122,154,189,226,267,312,361,414,471,532,597,666,739,816,897,982,1070,1017,932,851,774,701,630,699,772,849,930,1015,1095,1098,1016,927,842,761,684,611,542,477,416,359,306,257,212,171,134,101,72,47,67,90,115,144,177,214,255,300,349,402,459,520,585,654,727,804,885,970,1058,1029,944,861,942,1027,1101,1104,1028,939,854,773,696,623,554,489,428,371,318,269,224,183,146,113,84,59,38,21,8,16,27,40,57,78,103,132,165,202,243,288,337,390,447,508,573,642,715,792,873,958,1046,1109,1108,1041,953,868,787,710,637,568,503,442,385,332,283,238,197,160,127,98,73,52,35,22,13,6,11,20,33,50,71,96,125,158,195,236,281,330,383,440,501,566,635,708,785,866,951,1040,1044,952,863,947,1034,1053,965,880,799,722,649,580,515,454,397,344,295,250,209,172,139,110,85,64,45,62,83,108,137,170,207,248,293,342,395,452,513,578,647,720,797,878,963,1051,1056,964,875,790,709,632,704,779,856,937,1022,1065,977,892,811,734,661,592,527,466,409,356,307,262,221,184,151,120,149,182,219,260,305,354,407,464,525,590,659,732,809,890,975,1063,1068,976,887,802,721,644,571,502,437,497,560,625,694,767,844,925,1010,1077,989,904,823,746,673,604,539,478,421,368,319,274,231,272,317,366,419,476,537,602,671,744,821,902,987,1075,1080,988,899,814,733,656,583,514,449,388,331,278,326,377,430,487,548,613,682,755,832,913,998,1086,1001,916,835,758,685,616,551,490,433,378,431,488,549,614,683,756,833,914,999,1087,1090,1000,911,826,745,668,595,526,461,400,343,290,241,196,155,191,230,271,316,365,418,475,536,601,670,743,820,901,986,1074,1013,928,847,770,697,628,561,626,695,768,845,926,1011,1093,1096,1012,923,838,757,680,607,538,473,412,355,302,253,208,167,130,97,68,92,119,148,181,218,259,304,353,406,463,524,589,658,731,808,889,974,1062,1025,940,859,780,857,938,1023,1099,1102,1024,935,850,769,692,619,550,485,424,367,314,265,220,179,142,109,80,55,34,17,29,44,61,82,107,136,169,206,247,292,341,394,451,512,577,646,719,796,877,962,1050,1035,1105,1038,949,864,783,706,633,564,499,438,381,328,279,234,193,156,123,94,69,48,31,18,9,4)(1110,1111,1112)Polycyclic presentation:
2^2,3,1109|a^2=b,c^a=c^2,d^a=d^354,d^b=d^1108837 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13308, 7);