Finite Group DatabasebetaFinite Group Databasebeta| order | 13476 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| Fitting, socle | C2×C1123 ⋊ C2×C3 |
| derived, Sylow | C1123 ⋊ C22×C3 |
| Hall | C2×D1123 ⋊ C3 |
| Hall | C1123⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D1123⋊C3 |
| commutator, cosocle, hypocenter | C1123 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C1123 | C2×C3 |
| LCS | G ▷ C1123 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1125 |
| transitive degree | μt(G) = 2246 |
A minimal faithful permutation representation — 2 generators on 1125 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)(551,718,719,632,633,806)(553,716,717,635,631,710)(556,625,715,637,638,711)(563,729,730,644,645,706)(570,736,737,651,652,701)(577,743,744,658,659,696)(584,750,751,665,666,691)(591,757,758,672,673,686)(596,764,765,678,679,681)(601,770,762,683,684,676)(606,775,755,688,689,669)(611,780,748,693,694,662)(616,785,741,698,699,655)(620,790,734,703,704,648)(623,795,727,707,708,641)(634,811,812,721,722,898)(636,809,810,723,720,803)(639,714,808,724,725,804)(646,818,819,731,732,799)(653,823,824,738,739,794)(660,830,831,745,746,789)(667,837,838,752,753,784)(674,844,845,759,760,779)(680,851,852,766,767,774)(685,858,856,771,772,769)(690,863,849,776,777,763)(695,868,842,781,782,756)(700,873,835,786,787,749)(705,878,828,791,792,742)(709,883,821,796,797,735)(712,888,816,800,801,728)(726,807,899,813,814,896)(733,900,903,815,820,892)(740,906,907,825,826,887)(747,910,911,832,833,882)(754,917,918,839,840,877)(761,924,925,846,847,872)(768,931,932,853,854,867)(773,938,939,859,860,862)(778,944,936,864,865,857)(783,949,929,869,870,850)(788,954,922,874,875,843)(793,959,915,879,880,836)(798,964,908,884,885,829)(802,969,904,889,890,822)(805,974,901,893,894,817)(827,978,979,902,897,973)(834,982,983,912,913,968)(841,986,987,919,920,963)(848,990,991,926,927,958)(855,997,998,933,934,953)(861,1004,1005,940,941,948)(866,1011,1009,945,946,943)(871,1016,1002,950,951,937)(876,1021,995,955,956,930)(881,1026,988,960,961,923)(886,1031,984,965,966,916)(891,1036,980,970,971,909)(895,1040,914,975,976,905)(921,1041,1042,981,977,1035)(928,1045,1046,992,993,1030)(935,1049,1050,999,1000,1025)(942,1053,1054,1006,1007,1020)(947,1060,1061,1012,1013,1015)(952,1066,1058,1017,1018,1010)(957,1071,1051,1022,1023,1003)(962,1076,1047,1027,1028,996)(967,1081,1043,1032,1033,989)(972,1085,994,1037,1038,985)(1001,1086,1087,1044,1039,1080)(1008,1090,1091,1055,1056,1075)(1014,1094,1095,1062,1063,1070)(1019,1098,1096,1067,1068,1065)(1024,1103,1092,1072,1073,1059)(1029,1108,1088,1077,1078,1052)(1034,1112,1057,1082,1083,1048)(1064,1113,1114,1089,1084,1107)(1069,1117,1118,1099,1100,1102)(1074,1120,1115,1104,1105,1097)(1079,1121,1101,1109,1110,1093)(1106,1122,1123,1116,1111,1119);(1,2,5,11,19,32,51,74,103,138,179,226,279,338,403,474,551,634,723,635,552,475,404,339,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,343,408,479,556,639,726,815,731,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,728,817,902,825,738,651,568,491,420,355,296,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,293,352,417,488,565,648,735,822,905,912,832,745,658,575,498,427,362,303,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,300,359,424,495,572,655,742,829,909,981,919,839,752,665,582,505,434,368,308,254,206,163,107,50,71,84,88,91,93,125,162,204,252,306,366,431,502,579,662,749,836,916,985,992,926,846,759,672,589,511,439,373,313,259,210,142,73,100,114,119,123,126,128,166,209,257,311,371,437,509,586,669,756,843,923,989,1044,999,933,853,766,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,763,850,930,996,1048,1055,1006,940,859,771,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,769,857,937,1003,1052,1089,1062,1012,945,864,776,688,604,526,454,387,283,178,223,237,244,250,255,260,264,267,269,325,386,452,524,602,686,774,862,943,1010,1059,1093,1099,1067,1017,950,869,781,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,779,867,948,1015,1065,1097,1116,1104,1072,1022,955,874,786,698,614,535,407,278,335,349,356,363,369,374,379,384,388,391,393,461,534,612,696,784,872,953,1020,1070,1102,1119,1122,1109,1077,1027,960,879,791,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,789,877,958,1025,1075,1107,1121,1120,1117,1113,1082,1032,965,884,796,707,555,402,471,485,492,499,506,512,517,522,527,532,536,539,541,621,706,794,882,963,1030,1080,1112,1108,1103,1098,1094,1090,1086,1037,970,889,800,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,799,887,968,1035,1085,1081,1076,1071,1066,1060,1053,1049,1045,1041,975,893,725,550,631,645,652,659,666,673,679,684,689,694,699,704,708,711,713,803,892,973,1040,1036,1031,1026,1021,1016,1011,1004,997,990,986,982,978,814,633,720,732,739,746,753,760,767,772,777,782,787,792,797,801,804,806,812,899,974,969,964,959,954,949,944,938,931,924,917,910,906,900,722,811,820,826,833,840,847,854,860,865,870,875,880,885,890,894,896,898,719,808,888,883,878,873,868,863,858,851,844,837,830,823,818,809,718,807,897,913,920,927,934,941,946,951,956,961,966,971,976,979,903,810,630,715,795,790,785,780,775,770,764,757,750,743,736,729,716,629,714,805,895,977,993,1000,1007,1013,1018,1023,1028,1033,1038,1042,983,907,819,717,547,626,702,697,692,687,682,677,670,663,656,649,642,627,546,625,712,802,891,972,1039,1056,1063,1068,1073,1078,1083,1087,1046,987,911,824,730,628,470,543,613,608,603,598,593,587,580,573,566,559,544,469,542,623,709,798,886,967,1034,1084,1100,1105,1110,1114,1091,1050,991,918,831,737,643,545,399,466,530,525,520,515,510,503,496,489,482,467,398,465,540,620,705,793,881,962,1029,1079,1111,1123,1118,1095,1054,998,925,838,744,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,788,876,957,1024,1074,1106,1115,1096,1061,1005,932,845,751,657,567,483,397,275,330,382,377,372,367,360,353,346,331,274,329,392,460,533,611,695,783,871,952,1019,1069,1101,1092,1058,1009,939,852,758,664,574,490,412,332,222,271,317,312,307,301,294,287,272,221,270,327,389,456,528,606,690,778,866,947,1014,1064,1088,1051,1002,936,856,765,671,581,497,419,347,273,175,218,258,253,248,241,234,219,174,217,268,324,385,451,523,601,685,773,861,942,1008,1057,1047,995,929,849,762,675,588,504,426,354,288,220,134,171,205,200,194,187,172,133,170,215,265,320,380,446,518,596,680,768,855,935,1001,1043,988,922,842,755,668,585,508,433,361,295,235,173,99,130,158,153,146,131,98,129,168,212,261,315,375,441,513,591,674,761,848,928,994,984,915,835,748,661,578,501,430,365,302,242,188,132,70,95,117,111,96,69,94,127,165,208,256,310,370,436,507,584,667,754,841,921,980,908,828,741,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,747,834,914,904,821,734,647,564,487,416,351,292,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,298,357,422,493,570,653,740,827,901,816,727,640,557,480,409,344,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,291,350,415,486,563,646,733,813,724,637,554,477,406,341,282,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,281,340,405,476,553,636,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)(1124,1125)386 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13476, 8);