C₂²×C₁₁₁₇⋊C₃ #27211564

order	13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂²×C₁₁₁₇⋊C₃

Splittings

Fitting, socle, Hall	C₂²×C₁₁₁₇ ⋊ C₃
derived, Sylow	C₁₁₁₇ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	C₁₁₁₇⋊C₃
commutator, cosocle, hypocenter	C₁₁₁₇	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₁₁₇	C₃
LCS	G ▷ C₁₁₁₇
UCS	C₁ ◁ C₂²

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1121
transitive degree	μ_t(G) = 4468

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

Conjugacy classes & character table

1500 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13404, 15);

C22×C1117⋊C3 #27211564