Dic₄₅₉₃ #32913569

order	18372
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅₉₃
unambiguous semidirect product	C₃⋊Dic₁₅₃₁
unambiguous semidirect product	C₁₅₃₁⋊Dic₃

Splittings

derived, Hall	C₃×C₁₅₃₁ ⋊ C₄
Sylow	C₁₅₃₁ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₅₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₅₉₃
commutator, hypocenter	C₃×C₁₅₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₅₃₁	C₂
LCS	G ▷ C₃×C₁₅₃₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1538
transitive degree	μ_t(G) = 18372

A minimal faithful permutation representation — 2 generators on 1538 points:

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

Polycyclic presentation:

2,3,1531,2|a^2=d,b^a=b^2,c^a=c^1530

Conjugacy classes & character table

4596 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₁₈₆, Dic₁₃₇₇₉
maximal normal subgroup of	Dic₉₁₈₆
maximal quotient of	Dic₁₃₇₇₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18372, 6);

Dic4593 #32913569