Dic₆₈₃₅ #41032286

order	27340
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₈₃₅
unambiguous semidirect product	C₁₃₆₇⋊Dic₅

Splittings

derived, Hall	C₅×C₁₃₆₇ ⋊ C₄
Sylow	C₁₃₆₇ ⋊ Dic₅
Sylow	C₅ ⋊ Dic₁₃₆₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₈₃₅
commutator, hypocenter	C₅×C₁₃₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₃₆₇	C₂
LCS	G ▷ C₅×C₁₃₆₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1376
transitive degree	μ_t(G) = 27340

A minimal faithful permutation representation — 2 generators on 1376 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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(446,941)(447,940)(448,939)(449,938)(450,937)(451,936)(452,935)(453,934)(454,933)(455,932)(456,931)(457,930)(458,929)(459,928)(460,927)(461,926)(462,925)(463,924)(464,923)(465,922)(466,921)(467,920)(468,919)(469,918)(470,917)(471,916)(472,915)(473,914)(474,913)(475,912)(476,911)(477,910)(478,909)(479,908)(480,907)(481,906)(482,905)(483,904)(484,903)(485,902)(486,901)(487,900)(488,899)(489,898)(490,897)(491,896)(492,895)(493,894)(494,893)(495,892)(496,891)(497,890)(498,889)(499,888)(500,887)(501,886)(502,885)(503,884)(504,883)(505,882)(506,881)(507,880)(508,879)(509,878)(510,877)(511,876)(512,875)(513,874)(514,873)(515,872)(516,871)(517,870)(518,869)(519,868)(520,867)(521,866)(522,865)(523,864)(524,863)(525,862)(526,861)(527,860)(528,859)(529,858)(530,857)(531,856)(532,855)(533,854)(534,853)(535,852)(536,851)(537,850)(538,849)(539,848)(540,847)(541,846)(542,845)(543,844)(544,843)(545,842)(546,841)(547,840)(548,839)(549,838)(550,837)(551,836)(552,835)(553,834)(554,833)(555,832)(556,831)(557,830)(558,829)(559,828)(560,827)(561,826)(562,825)(563,824)(564,823)(565,822)(566,821)(567,820)(568,819)(569,818)(570,817)(571,816)(572,815)(573,814)(574,813)(575,812)(576,811)(577,810)(578,809)(579,808)(580,807)(581,806)(582,805)(583,804)(584,803)(585,802)(586,801)(587,800)(588,799)(589,798)(590,797)(591,796)(592,795)(593,794)(594,793)(595,792)(596,791)(597,790)(598,789)(599,788)(600,787)(601,786)(602,785)(603,784)(604,783)(605,782)(606,781)(607,780)(608,779)(609,778)(610,777)(611,776)(612,775)(613,774)(614,773)(615,772)(616,771)(617,770)(618,769)(619,768)(620,767)(621,766)(622,765)(623,764)(624,763)(625,762)(626,761)(627,760)(628,759)(629,758)(630,757)(631,756)(632,755)(633,754)(634,753)(635,752)(636,751)(637,750)(638,749)(639,748)(640,747)(641,746)(642,745)(643,744)(644,743)(645,742)(646,741)(647,740)(648,739)(649,738)(650,737)(651,736)(652,735)(653,734)(654,733)(655,732)(656,731)(657,730)(658,729)(659,728)(660,727)(661,726)(662,725)(663,724)(664,723)(665,722)(666,721)(667,720)(668,719)(669,718)(670,717)(671,716)(672,715)(673,714)(674,713)(675,712)(676,711)(677,710)(678,709)(679,708)(680,707)(681,706)(682,705)(683,704)(684,703)(685,702)(686,701)(687,700)(688,699)(689,698)(690,697)(691,696)(692,695)(693,694)

Polycyclic presentation:

2,5,1367,2|a^2=d,b^a=b^4,c^a=c^1366

Conjugacy classes & character table

6838 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₆₇₀
maximal normal subgroup of	Dic₁₃₆₇₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27340, 4);

Dic6835 #41032286