Dic₇₆₅₅ #44499516

order	30620
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₆₅₅
unambiguous semidirect product	C₁₅₃₁⋊Dic₅

Splittings

derived, Hall	C₅×C₁₅₃₁ ⋊ C₄
Sylow	C₁₅₃₁ ⋊ Dic₅
Sylow	C₅ ⋊ Dic₁₅₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₆₅₅
commutator, hypocenter	C₅×C₁₅₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₅₃₁	C₂
LCS	G ▷ C₅×C₁₅₃₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1540
transitive degree	μ_t(G) = 30620

A minimal faithful permutation representation — 2 generators on 1540 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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58,1193)(359,1192)(360,1191)(361,1190)(362,1189)(363,1188)(364,1187)(365,1186)(366,1185)(367,1184)(368,1183)(369,1182)(370,1181)(371,1180)(372,1179)(373,1178)(374,1177)(375,1176)(376,1175)(377,1174)(378,1173)(379,1172)(380,1171)(381,1170)(382,1169)(383,1168)(384,1167)(385,1166)(386,1165)(387,1164)(388,1163)(389,1162)(390,1161)(391,1160)(392,1159)(393,1158)(394,1157)(395,1156)(396,1155)(397,1154)(398,1153)(399,1152)(400,1151)(401,1150)(402,1149)(403,1148)(404,1147)(405,1146)(406,1145)(407,1144)(408,1143)(409,1142)(410,1141)(411,1140)(412,1139)(413,1138)(414,1137)(415,1136)(416,1135)(417,1134)(418,1133)(419,1132)(420,1131)(421,1130)(422,1129)(423,1128)(424,1127)(425,1126)(426,1125)(427,1124)(428,1123)(429,1122)(430,1121)(431,1120)(432,1119)(433,1118)(434,1117)(435,1116)(436,1115)(437,1114)(438,1113)(439,1112)(440,1111)(441,1110)(442,1109)(443,1108)(444,1107)(445,1106)(446,1105)(447,1104)(448,1103)(449,1102)(450,1101)(451,1100)(452,1099)(453,1098)(454,1097)(455,1096)(456,1095)(457,1094)(458,1093)(459,1092)(460,1091)(461,1090)(462,1089)(463,1088)(464,1087)(465,1086)(466,1085)(467,1084)(468,1083)(469,1082)(470,1081)(471,1080)(472,1079)(473,1078)(474,1077)(475,1076)(476,1075)(477,1074)(478,1073)(479,1072)(480,1071)(481,1070)(482,1069)(483,1068)(484,1067)(485,1066)(486,1065)(487,1064)(488,1063)(489,1062)(490,1061)(491,1060)(492,1059)(493,1058)(494,1057)(495,1056)(496,1055)(497,1054)(498,1053)(499,1052)(500,1051)(501,1050)(502,1049)(503,1048)(504,1047)(505,1046)(506,1045)(507,1044)(508,1043)(509,1042)(510,1041)(511,1040)(512,1039)(513,1038)(514,1037)(515,1036)(516,1035)(517,1034)(518,1033)(519,1032)(520,1031)(521,1030)(522,1029)(523,1028)(524,1027)(525,1026)(526,1025)(527,1024)(528,1023)(529,1022)(530,1021)(531,1020)(532,1019)(533,1018)(534,1017)(535,1016)(536,1015)(537,1014)(538,1013)(539,1012)(540,1011)(541,1010)(542,1009)(543,1008)(544,1007)(545,1006)(546,1005)(547,1004)(548,1003)(549,1002)(550,1001)(551,1000)(552,999)(553,998)(554,997)(555,996)(556,995)(557,994)(558,993)(559,992)(560,991)(561,990)(562,989)(563,988)(564,987)(565,986)(566,985)(567,984)(568,983)(569,982)(570,981)(571,980)(572,979)(573,978)(574,977)(575,976)(576,975)(577,974)(578,973)(579,972)(580,971)(581,970)(582,969)(583,968)(584,967)(585,966)(586,965)(587,964)(588,963)(589,962)(590,961)(591,960)(592,959)(593,958)(594,957)(595,956)(596,955)(597,954)(598,953)(599,952)(600,951)(601,950)(602,949)(603,948)(604,947)(605,946)(606,945)(607,944)(608,943)(609,942)(610,941)(611,940)(612,939)(613,938)(614,937)(615,936)(616,935)(617,934)(618,933)(619,932)(620,931)(621,930)(622,929)(623,928)(624,927)(625,926)(626,925)(627,924)(628,923)(629,922)(630,921)(631,920)(632,919)(633,918)(634,917)(635,916)(636,915)(637,914)(638,913)(639,912)(640,911)(641,910)(642,909)(643,908)(644,907)(645,906)(646,905)(647,904)(648,903)(649,902)(650,901)(651,900)(652,899)(653,898)(654,897)(655,896)(656,895)(657,894)(658,893)(659,892)(660,891)(661,890)(662,889)(663,888)(664,887)(665,886)(666,885)(667,884)(668,883)(669,882)(670,881)(671,880)(672,879)(673,878)(674,877)(675,876)(676,875)(677,874)(678,873)(679,872)(680,871)(681,870)(682,869)(683,868)(684,867)(685,866)(686,865)(687,864)(688,863)(689,862)(690,861)(691,860)(692,859)(693,858)(694,857)(695,856)(696,855)(697,854)(698,853)(699,852)(700,851)(701,850)(702,849)(703,848)(704,847)(705,846)(706,845)(707,844)(708,843)(709,842)(710,841)(711,840)(712,839)(713,838)(714,837)(715,836)(716,835)(717,834)(718,833)(719,832)(720,831)(721,830)(722,829)(723,828)(724,827)(725,826)(726,825)(727,824)(728,823)(729,822)(730,821)(731,820)(732,819)(733,818)(734,817)(735,816)(736,815)(737,814)(738,813)(739,812)(740,811)(741,810)(742,809)(743,808)(744,807)(745,806)(746,805)(747,804)(748,803)(749,802)(750,801)(751,800)(752,799)(753,798)(754,797)(755,796)(756,795)(757,794)(758,793)(759,792)(760,791)(761,790)(762,789)(763,788)(764,787)(765,786)(766,785)(767,784)(768,783)(769,782)(770,781)(771,780)(772,779)(773,778)(774,777)(775,776)

Polycyclic presentation:

2,5,1531,2|a^2=d,b^a=b^4,c^a=c^1530

Conjugacy classes & character table

7658 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₃₁₀
maximal normal subgroup of	Dic₁₅₃₁₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30620, 6);

Dic7655 #44499516