D₁₆₀₀₃ #44544932

order	32006
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₀₀₃
unambiguous semidirect product	C₁₃⋊D₁₂₃₁
unambiguous semidirect product	C₁₂₃₁⋊D₁₃

Splittings

Fitting, socle, derived, Hall	C₁₃×C₁₂₃₁ ⋊ C₂
Sylow	C₁₂₃₁ ⋊ D₁₃
Sylow	C₁₃ ⋊ D₁₂₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₃×C₁₂₃₁	C₂
LCS	G ▷ C₁₃×C₁₂₃₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1244
transitive degree	μ_t(G) = 16003

A minimal faithful permutation representation — 2 generators on 1244 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223,1224,1225,1226,1227,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,1238,1239,1240,1241,1242,1243,1244);(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,1244)(16,1243)(17,1242)(18,1241)(19,1240)(20,1239)(21,1238)(22,1237)(23,1236)(24,1235)(25,1234)(26,1233)(27,1232)(28,1231)(29,1230)(30,1229)(31,1228)(32,1227)(33,1226)(34,1225)(35,1224)(36,1223)(37,1222)(38,1221)(39,1220)(40,1219)(41,1218)(42,1217)(43,1216)(44,1215)(45,1214)(46,1213)(47,1212)(48,1211)(49,1210)(50,1209)(51,1208)(52,1207)(53,1206)(54,1205)(55,1204)(56,1203)(57,1202)(58,1201)(59,1200)(60,1199)(61,1198)(62,1197)(63,1196)(64,1195)(65,1194)(66,1193)(67,1192)(68,1191)(69,1190)(70,1189)(71,1188)(72,1187)(73,1186)(74,1185)(75,1184)(76,1183)(77,1182)(78,1181)(79,1180)(80,1179)(81,1178)(82,1177)(83,1176)(84,1175)(85,1174)(86,1173)(87,1172)(88,1171)(89,1170)(90,1169)(91,1168)(92,1167)(93,1166)(94,1165)(95,1164)(96,1163)(97,1162)(98,1161)(99,1160)(100,1159)(101,1158)(102,1157)(103,1156)(104,1155)(105,1154)(106,1153)(107,1152)(108,1151)(109,1150)(110,1149)(111,1148)(112,1147)(113,1146)(114,1145)(115,1144)(116,1143)(117,1142)(118,1141)(119,1140)(120,1139)(121,1138)(122,1137)(123,1136)(124,1135)(125,1134)(126,1133)(127,1132)(128,1131)(129,1130)(130,1129)(131,1128)(132,1127)(133,1126)(134,1125)(135,1124)(136,1123)(137,1122)(138,1121)(139,1120)(140,1119)(141,1118)(142,1117)(143,1116)(144,1115)(145,1114)(146,1113)(147,1112)(148,1111)(149,1110)(150,1109)(151,1108)(152,1107)(153,1106)(154,1105)(155,1104)(156,1103)(157,1102)(158,1101)(159,1100)(160,1099)(161,1098)(162,1097)(163,1096)(164,1095)(165,1094)(166,1093)(167,1092)(168,1091)(169,1090)(170,1089)(171,1088)(172,1087)(173,1086)(174,1085)(175,1084)(176,1083)(177,1082)(178,1081)(179,1080)(180,1079)(181,1078)(182,1077)(183,1076)(184,1075)(185,1074)(186,1073)(187,1072)(188,1071)(189,1070)(190,1069)(191,1068)(192,1067)(193,1066)(194,1065)(195,1064)(196,1063)(197,1062)(198,1061)(199,1060)(200,1059)(201,1058)(202,1057)(203,1056)(204,1055)(205,1054)(206,1053)(207,1052)(208,1051)(209,1050)(210,1049)(211,1048)(212,1047)(213,1046)(214,1045)(215,1044)(216,1043)(217,1042)(218,1041)(219,1040)(220,1039)(221,1038)(222,1037)(223,1036)(224,1035)(225,1034)(226,1033)(227,1032)(228,1031)(229,1030)(230,1029)(231,1028)(232,1027)(233,1026)(234,1025)(235,1024)(236,1023)(237,1022)(238,1021)(239,1020)(240,1019)(241,1018)(242,1017)(243,1016)(244,1015)(245,1014)(246,1013)(247,1012)(248,1011)(249,1010)(250,1009)(251,1008)(252,1007)(253,1006)(254,1005)(255,1004)(256,1003)(257,1002)(258,1001)(259,1000)(260,999)(261,998)(262,997)(263,996)(264,995)(265,994)(266,993)(267,992)(268,991)(269,990)(270,989)(271,988)(272,987)(273,986)(274,985)(275,984)(276,983)(277,982)(278,981)(279,980)(280,979)(281,978)(282,977)(283,976)(284,975)(285,974)(286,973)(287,972)(288,971)(289,970)(290,969)(291,968)(292,967)(293,966)(294,965)(295,964)(296,963)(297,962)(298,961)(299,960)(300,959)(301,958)(302,957)(303,956)(304,955)(305,954)(306,953)(307,952)(308,951)(309,950)(310,949)(311,948)(312,947)(313,946)(314,945)(315,944)(316,943)(317,942)(318,941)(319,940)(320,939)(321,938)(322,937)(323,936)(324,935)(325,934)(326,933)(327,932)(328,931)(329,930)(330,929)(331,928)(332,927)(333,926)(334,925)(335,924)(336,923)(337,922)(338,921)(339,920)(340,919)(341,918)(342,917)(343,916)(344,915)(345,914)(346,913)(347,912)(348,911)(349,910)(350,909)(351,908)(352,907)(353,906)(354,905)(355,904)(356,903)(357,902)(358,901)(359,900)(360,899)(361,898)(362,897)(363,896)(364,895)(365,894)(366,893)(367,892)(368,891)(369,890)(370,889)(371,888)(372,887)(373,886)(374,885)(375,884)(376,883)(377,882)(378,881)(379,880)(380,879)(381,878)(382,877)(383,876)(384,875)(385,874)(386,873)(387,872)(388,871)(389,870)(390,869)(391,868)(392,867)(393,866)(394,865)(395,864)(396,863)(397,862)(398,861)(399,860)(400,859)(401,858)(402,857)(403,856)(404,855)(405,854)(406,853)(407,852)(408,851)(409,850)(410,849)(411,848)(412,847)(413,846)(414,845)(415,844)(416,843)(417,842)(418,841)(419,840)(420,839)(421,838)(422,837)(423,836)(424,835)(425,834)(426,833)(427,832)(428,831)(429,830)(430,829)(431,828)(432,827)(433,826)(434,825)(435,824)(436,823)(437,822)(438,821)(439,820)(440,819)(441,818)(442,817)(443,816)(444,815)(445,814)(446,813)(447,812)(448,811)(449,810)(450,809)(451,808)(452,807)(453,806)(454,805)(455,804)(456,803)(457,802)(458,801)(459,800)(460,799)(461,798)(462,797)(463,796)(464,795)(465,794)(466,793)(467,792)(468,791)(469,790)(470,789)(471,788)(472,787)(473,786)(474,785)(475,784)(476,783)(477,782)(478,781)(479,780)(480,779)(481,778)(482,777)(483,776)(484,775)(485,774)(486,773)(487,772)(488,771)(489,770)(490,769)(491,768)(492,767)(493,766)(494,765)(495,764)(496,763)(497,762)(498,761)(499,760)(500,759)(501,758)(502,757)(503,756)(504,755)(505,754)(506,753)(507,752)(508,751)(509,750)(510,749)(511,748)(512,747)(513,746)(514,745)(515,744)(516,743)(517,742)(518,741)(519,740)(520,739)(521,738)(522,737)(523,736)(524,735)(525,734)(526,733)(527,732)(528,731)(529,730)(530,729)(531,728)(532,727)(533,726)(534,725)(535,724)(536,723)(537,722)(538,721)(539,720)(540,719)(541,718)(542,717)(543,716)(544,715)(545,714)(546,713)(547,712)(548,711)(549,710)(550,709)(551,708)(552,707)(553,706)(554,705)(555,704)(556,703)(557,702)(558,701)(559,700)(560,699)(561,698)(562,697)(563,696)(564,695)(565,694)(566,693)(567,692)(568,691)(569,690)(570,689)(571,688)(572,687)(573,686)(574,685)(575,684)(576,683)(577,682)(578,681)(579,680)(580,679)(581,678)(582,677)(583,676)(584,675)(585,674)(586,673)(587,672)(588,671)(589,670)(590,669)(591,668)(592,667)(593,666)(594,665)(595,664)(596,663)(597,662)(598,661)(599,660)(600,659)(601,658)(602,657)(603,656)(604,655)(605,654)(606,653)(607,652)(608,651)(609,650)(610,649)(611,648)(612,647)(613,646)(614,645)(615,644)(616,643)(617,642)(618,641)(619,640)(620,639)(621,638)(622,637)(623,636)(624,635)(625,634)(626,633)(627,632)(628,631)(629,630)

Polycyclic presentation:

2,13,1231|b^a=b^12,c^a=c^1230

Conjugacy classes & character table

8003 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₀₀₃
maximal normal subgroup of	C₂×D₁₆₀₀₃
maximal quotient of	Dic₁₆₀₀₃, C₂×D₁₆₀₀₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32006, 3);

D16003 #44544932