Finite Group DatabasebetaFinite Group Databasebeta| order | 13548 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| Fitting, socle | C2×C1129 ⋊ C2×C3 |
| derived, Sylow | C1129 ⋊ C22×C3 |
| Hall | C2×D1129 ⋊ C3 |
| Hall | C1129⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1129⋊C3 |
| commutator, cosocle, hypocenter | C1129 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C1129 | C2×C3 |
| LCS | G ▷ C1129 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1131 |
| transitive degree | μt(G) = 2258 |
A minimal faithful permutation representation — 2 generators on 1131 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,64)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,69,70,49,50,93)(34,67,68,52,48,61)(37,42,66,54,55,62)(40,82,80,58,59,57)(51,98,99,72,73,128)(53,96,97,75,71,90)(56,65,95,77,78,91)(60,111,112,83,84,86)(63,117,109,87,88,81)(74,133,134,101,102,169)(76,131,132,104,100,125)(79,94,130,106,107,126)(85,146,147,113,114,121)(89,153,151,118,119,116)(92,158,144,122,123,110)(103,174,175,136,137,216)(105,172,173,139,135,166)(108,129,171,141,142,167)(115,187,188,148,149,162)(120,194,195,154,155,157)(124,200,192,159,160,152)(127,205,185,163,164,145)(138,221,222,177,178,269)(140,219,220,180,176,213)(143,170,218,182,183,214)(150,234,235,189,190,209)(156,241,242,196,197,204)(161,248,246,201,202,199)(165,253,239,206,207,193)(168,258,232,210,211,186)(179,274,275,224,225,328)(181,272,273,227,223,266)(184,217,271,229,230,267)(191,287,288,236,237,262)(198,294,295,243,244,257)(203,301,302,249,250,252)(208,307,299,254,255,247)(212,312,292,259,260,240)(215,317,285,263,264,233)(226,333,334,277,278,393)(228,331,332,280,276,325)(231,270,330,282,283,326)(238,346,347,289,290,321)(245,353,354,296,297,316)(251,360,361,303,304,311)(256,367,365,308,309,306)(261,372,358,313,314,300)(265,377,351,318,319,293)(268,382,344,322,323,286)(279,398,399,336,337,464)(281,396,397,339,335,390)(284,329,395,341,342,391)(291,411,412,348,349,386)(298,418,419,355,356,381)(305,425,426,362,363,376)(310,432,433,368,369,371)(315,438,430,373,374,366)(320,443,423,378,379,359)(324,448,416,383,384,352)(327,453,409,387,388,345)(338,469,470,401,402,541)(340,467,468,404,400,461)(343,394,466,406,407,462)(350,482,483,413,414,457)(357,489,490,420,421,452)(364,496,497,427,428,447)(370,503,504,434,435,442)(375,510,508,439,440,437)(380,515,501,444,445,431)(385,520,494,449,450,424)(389,525,487,454,455,417)(392,530,480,458,459,410)(403,546,547,472,473,624)(405,544,545,475,471,538)(408,465,543,477,478,539)(415,559,560,484,485,534)(422,566,567,491,492,529)(429,573,574,498,499,524)(436,580,581,505,506,519)(441,587,588,511,512,514)(446,593,585,516,517,509)(451,598,578,521,522,502)(456,603,571,526,527,495)(460,608,564,531,532,488)(463,613,557,535,536,481)(474,629,630,549,550,713)(476,627,628,552,548,621)(479,542,626,554,555,622)(486,642,643,561,562,617)(493,649,650,568,569,612)(500,656,657,575,576,607)(507,663,664,582,583,602)(513,670,671,589,590,597)(518,677,675,594,595,592)(523,682,668,599,600,586)(528,687,661,604,605,579)(533,692,654,609,610,572)(537,697,647,614,615,565)(540,702,640,618,619,558)(551,718,719,632,633,808)(553,716,717,635,631,710)(556,625,715,637,638,711)(563,731,732,644,645,706)(570,738,739,651,652,701)(577,745,746,658,659,696)(584,752,753,665,666,691)(591,759,760,672,673,686)(596,766,767,678,679,681)(601,772,764,683,684,676)(606,777,757,688,689,669)(611,782,750,693,694,662)(616,787,743,698,699,655)(620,792,736,703,704,648)(623,797,729,707,708,641)(634,813,814,721,722,903)(636,811,812,724,720,805)(639,714,810,726,727,806)(646,824,825,733,734,801)(653,829,830,740,741,796)(660,834,835,747,748,791)(667,841,842,754,755,786)(674,848,849,761,762,781)(680,855,856,768,769,776)(685,862,860,773,774,771)(690,867,853,778,779,765)(695,872,846,783,784,758)(700,877,839,788,789,751)(705,882,832,793,794,744)(709,887,827,798,799,737)(712,892,822,802,803,730)(723,831,891,742,816,915)(725,906,907,817,815,900)(728,809,905,819,820,901)(735,908,912,818,826,896)(749,918,919,836,837,886)(756,922,923,843,844,881)(763,929,930,850,851,876)(770,936,937,857,858,871)(775,943,944,863,864,866)(780,949,941,868,869,861)(785,954,934,873,874,854)(790,959,927,878,879,847)(795,964,920,883,884,840)(800,969,916,888,889,833)(804,974,913,893,894,828)(807,978,910,897,898,823)(821,904,973,838,909,985)(845,992,993,924,925,968)(852,996,997,931,932,963)(859,1000,1001,938,939,958)(865,1007,1008,945,946,953)(870,1014,1012,950,951,948)(875,1019,1005,955,956,942)(880,1024,998,960,961,935)(885,1029,994,965,966,928)(890,1034,990,970,971,921)(895,1039,988,975,976,917)(899,1043,986,979,980,914)(902,1038,926,982,983,911)(933,1047,1048,987,984,1033)(940,1055,1056,1002,1003,1028)(947,1059,1060,1009,1010,1023)(952,1063,1064,1015,1016,1018)(957,1069,1061,1020,1021,1013)(962,1074,1057,1025,1026,1006)(967,1079,1053,1030,1031,999)(972,1084,1051,1035,1036,995)(977,1088,1049,1040,1041,991)(981,1083,1004,1044,1045,989)(1011,1092,1093,1050,1046,1078)(1017,1100,1101,1065,1066,1073)(1022,1104,1102,1070,1071,1068)(1027,1106,1098,1075,1076,1062)(1032,1111,1096,1080,1081,1058)(1037,1115,1094,1085,1086,1054)(1042,1110,1067,1089,1090,1052)(1072,1119,1120,1095,1091,1105)(1077,1126,1123,1107,1108,1103)(1082,1127,1121,1112,1113,1099)(1087,1125,1109,1116,1117,1097)(1114,1128,1129,1122,1118,1124);(1,2,5,11,19,32,51,74,103,138,179,226,279,338,403,474,551,634,723,740,651,568,491,420,355,296,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,293,352,417,488,565,648,737,828,914,987,931,850,761,672,589,511,439,373,313,259,210,142,73,100,114,119,123,126,128,166,209,257,311,371,437,509,586,669,758,847,928,995,1052,1065,1015,950,868,778,688,604,526,454,387,283,178,223,237,244,250,255,260,264,267,269,325,386,452,524,602,686,776,866,948,1013,1062,1099,1122,1112,1080,1030,965,883,793,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,791,881,963,1028,1078,1110,1115,1111,1106,1104,1100,1092,1044,979,897,727,550,631,645,652,659,666,673,679,684,689,694,699,704,708,711,713,805,896,814,905,978,974,969,964,959,954,949,943,936,929,922,918,816,908,906,813,904,925,932,939,946,951,956,961,966,971,976,980,983,985,915,825,717,547,626,702,697,692,687,682,677,670,663,656,649,642,627,546,625,712,804,895,977,1042,1091,1108,1113,1117,1120,1101,1060,1001,930,842,746,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,790,880,962,1027,1077,1109,1096,1057,1005,941,860,767,671,581,497,419,347,273,175,218,258,253,248,241,234,219,174,217,268,324,385,451,523,601,685,775,865,947,1011,1049,990,920,839,750,661,578,501,430,365,302,242,188,132,70,95,117,111,96,69,94,127,165,208,256,310,370,436,507,584,667,756,845,926,910,822,729,640,557,480,409,344,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,291,350,415,486,563,646,735,817,724,635,552,475,404,339,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,343,408,479,556,639,728,821,836,747,658,575,498,427,362,303,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,300,359,424,495,572,655,744,833,917,989,1002,938,857,768,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,765,854,935,999,1054,1095,1070,1020,955,873,783,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,781,871,953,1018,1068,1103,1124,1128,1116,1085,1035,970,888,798,707,555,402,471,485,492,499,506,512,517,522,527,532,536,539,541,621,706,796,886,968,1033,1083,1088,1084,1079,1074,1069,1063,1059,1055,1047,982,820,633,720,734,741,748,755,762,769,774,779,784,789,794,799,803,806,808,900,907,719,810,892,887,882,877,872,867,862,855,848,841,834,829,824,811,718,809,902,984,1003,1010,1016,1021,1026,1031,1036,1041,1045,1048,993,919,830,732,628,470,543,613,608,603,598,593,587,580,573,566,559,544,469,542,623,709,800,890,972,1037,1087,1118,1129,1123,1102,1064,1008,937,849,753,657,567,483,397,275,330,382,377,372,367,360,353,346,331,274,329,392,460,533,611,695,785,875,957,1022,1072,1094,1053,998,934,853,764,675,588,504,426,354,288,220,134,171,205,200,194,187,172,133,170,215,265,320,380,446,518,596,680,770,859,940,1004,988,916,832,743,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,749,838,819,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,281,340,405,476,553,636,725,818,733,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,730,823,911,924,843,754,665,582,505,434,368,308,254,206,163,107,50,71,84,88,91,93,125,162,204,252,306,366,431,502,579,662,751,840,921,991,1050,1009,945,863,773,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,771,861,942,1006,1058,1097,1107,1075,1025,960,878,788,698,614,535,407,278,335,349,356,363,369,374,379,384,388,391,393,461,534,612,696,786,876,958,1023,1073,1105,1125,1127,1126,1119,1089,1040,975,893,802,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,801,891,973,1038,1043,1039,1034,1029,1024,1019,1014,1007,1000,996,992,909,722,815,826,831,837,844,851,858,864,869,874,879,884,889,894,898,901,903,912,812,630,715,797,792,787,782,777,772,766,759,752,745,738,731,716,629,714,807,899,981,1046,1066,1071,1076,1081,1086,1090,1093,1056,997,923,835,739,643,545,399,466,530,525,520,515,510,503,496,489,482,467,398,465,540,620,705,795,885,967,1032,1082,1114,1121,1098,1061,1012,944,856,760,664,574,490,412,332,222,271,317,312,307,301,294,287,272,221,270,327,389,456,528,606,690,780,870,952,1017,1067,1051,994,927,846,757,668,585,508,433,361,295,235,173,99,130,158,153,146,131,98,129,168,212,261,315,375,441,513,591,674,763,852,933,986,913,827,736,647,564,487,416,351,292,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,298,357,422,493,570,653,742,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)(1130,1131)388 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13548, 9);