C₂×D₆₀₀₅ #37562259

order	24020
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₂₀₁₀
direct product	C₂×D₆₀₀₅
unambiguous semidirect product	(C₂×C₅)⋊D₁₂₀₁
unambiguous semidirect product	(C₂×C₁₂₀₁)⋊D₅

Splittings

Fitting, socle	C₂×C₅×C₁₂₀₁ ⋊ C₂
derived, Hall	C₅×C₁₂₀₁ ⋊ C₂²
Sylow	C₁₂₀₁ ⋊ C₂×D₅
Sylow	C₅ ⋊ C₂×D₁₂₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₀₀₅
commutator, cosocle, hypocenter	C₅×C₁₂₀₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₅×C₁₂₀₁	C₂
LCS	G ▷ C₅×C₁₂₀₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1208
transitive degree	μ_t(G) = 12010

A minimal faithful permutation representation — 2 generators on 1208 points:

(1,2)(3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208);(4,7)(5,6)(9,1208)(10,1207)(11,1206)(12,1205)(13,1204)(14,1203)(15,1202)(16,1201)(17,1200)(18,1199)(19,1198)(20,1197)(21,1196)(22,1195)(23,1194)(24,1193)(25,1192)(26,1191)(27,1190)(28,1189)(29,1188)(30,1187)(31,1186)(32,1185)(33,1184)(34,1183)(35,1182)(36,1181)(37,1180)(38,1179)(39,1178)(40,1177)(41,1176)(42,1175)(43,1174)(44,1173)(45,1172)(46,1171)(47,1170)(48,1169)(49,1168)(50,1167)(51,1166)(52,1165)(53,1164)(54,1163)(55,1162)(56,1161)(57,1160)(58,1159)(59,1158)(60,1157)(61,1156)(62,1155)(63,1154)(64,1153)(65,1152)(66,1151)(67,1150)(68,1149)(69,1148)(70,1147)(71,1146)(72,1145)(73,1144)(74,1143)(75,1142)(76,1141)(77,1140)(78,1139)(79,1138)(80,1137)(81,1136)(82,1135)(83,1134)(84,1133)(85,1132)(86,1131)(87,1130)(88,1129)(89,1128)(90,1127)(91,1126)(92,1125)(93,1124)(94,1123)(95,1122)(96,1121)(97,1120)(98,1119)(99,1118)(100,1117)(101,1116)(102,1115)(103,1114)(104,1113)(105,1112)(106,1111)(107,1110)(108,1109)(109,1108)(110,1107)(111,1106)(112,1105)(113,1104)(114,1103)(115,1102)(116,1101)(117,1100)(118,1099)(119,1098)(120,1097)(121,1096)(122,1095)(123,1094)(124,1093)(125,1092)(126,1091)(127,1090)(128,1089)(129,1088)(130,1087)(131,1086)(132,1085)(133,1084)(134,1083)(135,1082)(136,1081)(137,1080)(138,1079)(139,1078)(140,1077)(141,1076)(142,1075)(143,1074)(144,1073)(145,1072)(146,1071)(147,1070)(148,1069)(149,1068)(150,1067)(151,1066)(152,1065)(153,1064)(154,1063)(155,1062)(156,1061)(157,1060)(158,1059)(159,1058)(160,1057)(161,1056)(162,1055)(163,1054)(164,1053)(165,1052)(166,1051)(167,1050)(168,1049)(169,1048)(170,1047)(171,1046)(172,1045)(173,1044)(174,1043)(175,1042)(176,1041)(177,1040)(178,1039)(179,1038)(180,1037)(181,1036)(182,1035)(183,1034)(184,1033)(185,1032)(186,1031)(187,1030)(188,1029)(189,1028)(190,1027)(191,1026)(192,1025)(193,1024)(194,1023)(195,1022)(196,1021)(197,1020)(198,1019)(199,1018)(200,1017)(201,1016)(202,1015)(203,1014)(204,1013)(205,1012)(206,1011)(207,1010)(208,1009)(209,1008)(210,1007)(211,1006)(212,1005)(213,1004)(214,1003)(215,1002)(216,1001)(217,1000)(218,999)(219,998)(220,997)(221,996)(222,995)(223,994)(224,993)(225,992)(226,991)(227,990)(228,989)(229,988)(230,987)(231,986)(232,985)(233,984)(234,983)(235,982)(236,981)(237,980)(238,979)(239,978)(240,977)(241,976)(242,975)(243,974)(244,973)(245,972)(246,971)(247,970)(248,969)(249,968)(250,967)(251,966)(252,965)(253,964)(254,963)(255,962)(256,961)(257,960)(258,959)(259,958)(260,957)(261,956)(262,955)(263,954)(264,953)(265,952)(266,951)(267,950)(268,949)(269,948)(270,947)(271,946)(272,945)(273,944)(274,943)(275,942)(276,941)(277,940)(278,939)(279,938)(280,937)(281,936)(282,935)(283,934)(284,933)(285,932)(286,931)(287,930)(288,929)(289,928)(290,927)(291,926)(292,925)(293,924)(294,923)(295,922)(296,921)(297,920)(298,919)(299,918)(300,917)(301,916)(302,915)(303,914)(304,913)(305,912)(306,911)(307,910)(308,909)(309,908)(310,907)(311,906)(312,905)(313,904)(314,903)(315,902)(316,901)(317,900)(318,899)(319,898)(320,897)(321,896)(322,895)(323,894)(324,893)(325,892)(326,891)(327,890)(328,889)(329,888)(330,887)(331,886)(332,885)(333,884)(334,883)(335,882)(336,881)(337,880)(338,879)(339,878)(340,877)(341,876)(342,875)(343,874)(344,873)(345,872)(346,871)(347,870)(348,869)(349,868)(350,867)(351,866)(352,865)(353,864)(354,863)(355,862)(356,861)(357,860)(358,859)(359,858)(360,857)(361,856)(362,855)(363,854)(364,853)(365,852)(366,851)(367,850)(368,849)(369,848)(370,847)(371,846)(372,845)(373,844)(374,843)(375,842)(376,841)(377,840)(378,839)(379,838)(380,837)(381,836)(382,835)(383,834)(384,833)(385,832)(386,831)(387,830)(388,829)(389,828)(390,827)(391,826)(392,825)(393,824)(394,823)(395,822)(396,821)(397,820)(398,819)(399,818)(400,817)(401,816)(402,815)(403,814)(404,813)(405,812)(406,811)(407,810)(408,809)(409,808)(410,807)(411,806)(412,805)(413,804)(414,803)(415,802)(416,801)(417,800)(418,799)(419,798)(420,797)(421,796)(422,795)(423,794)(424,793)(425,792)(426,791)(427,790)(428,789)(429,788)(430,787)(431,786)(432,785)(433,784)(434,783)(435,782)(436,781)(437,780)(438,779)(439,778)(440,777)(441,776)(442,775)(443,774)(444,773)(445,772)(446,771)(447,770)(448,769)(449,768)(450,767)(451,766)(452,765)(453,764)(454,763)(455,762)(456,761)(457,760)(458,759)(459,758)(460,757)(461,756)(462,755)(463,754)(464,753)(465,752)(466,751)(467,750)(468,749)(469,748)(470,747)(471,746)(472,745)(473,744)(474,743)(475,742)(476,741)(477,740)(478,739)(479,738)(480,737)(481,736)(482,735)(483,734)(484,733)(485,732)(486,731)(487,730)(488,729)(489,728)(490,727)(491,726)(492,725)(493,724)(494,723)(495,722)(496,721)(497,720)(498,719)(499,718)(500,717)(501,716)(502,715)(503,714)(504,713)(505,712)(506,711)(507,710)(508,709)(509,708)(510,707)(511,706)(512,705)(513,704)(514,703)(515,702)(516,701)(517,700)(518,699)(519,698)(520,697)(521,696)(522,695)(523,694)(524,693)(525,692)(526,691)(527,690)(528,689)(529,688)(530,687)(531,686)(532,685)(533,684)(534,683)(535,682)(536,681)(537,680)(538,679)(539,678)(540,677)(541,676)(542,675)(543,674)(544,673)(545,672)(546,671)(547,670)(548,669)(549,668)(550,667)(551,666)(552,665)(553,664)(554,663)(555,662)(556,661)(557,660)(558,659)(559,658)(560,657)(561,656)(562,655)(563,654)(564,653)(565,652)(566,651)(567,650)(568,649)(569,648)(570,647)(571,646)(572,645)(573,644)(574,643)(575,642)(576,641)(577,640)(578,639)(579,638)(580,637)(581,636)(582,635)(583,634)(584,633)(585,632)(586,631)(587,630)(588,629)(589,628)(590,627)(591,626)(592,625)(593,624)(594,623)(595,622)(596,621)(597,620)(598,619)(599,618)(600,617)(601,616)(602,615)(603,614)(604,613)(605,612)(606,611)(607,610)(608,609)

Conjugacy classes & character table

6008 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₄₀₂₀
maximal normal subgroup of	D₂₄₀₂₀
maximal quotient of	Dic₁₂₀₁₀, D₂₄₀₂₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24020, 20);

C2×D6005 #37562259