Finite Group Databasebeta

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C2×D16305 #72303646

order65220
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD32610
direct productC2×D16305

Splittings

Fitting, socleC2×C3×C5×C1087 ⋊ C2
derived, HallC3×C5×C1087 ⋊ C22
HallC5×C1087 ⋊ C2×D3
HallC3×C1087 ⋊ C2×D5
SylowC1087 ⋊ C2×D15
HallC3×C5 ⋊ C2×D1087
SylowC5 ⋊ C2×D3261
SylowC3 ⋊ C2×D5435

Subgroups

subgroupG/H
center, hypercenterC2D16305
commutator, cosocle, hypocenterC3×C5×C1087C22
FrattiniC1G
Fitting, socleC2×C3×C5×C1087C2
LCSG ▷ C3×C5×C1087
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 1097
transitive degreeμt(G) = 32610

A minimal faithful permutation representation — 2 generators on 1097 points:

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

Conjugacy classes & character table

16308 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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