Finite Group Databasebeta

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Dic12288 #72283913

order49152
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

dicyclicDic12288

Splittings

SylowC3 ⋊ Q16384

Subgroups

subgroupG/H
centerC2D12288
commutator, cosocleC3×C4096C22
FrattiniC4096C2×D3
FittingC3×C8192C2
socle, prime radicalC2×C3D4096
hypercenterC8192D3
hypocenterC3Q16384
LCSG ▷ C3×C4096 ▷ C3×C2048 ▷ C3×C1024 ▷ C3×C512 ▷ C3×C256 ▷ C3×C128 ▷ C3×C64 ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ C128 ◁ C256 ◁ C512 ◁ C1024 ◁ C2048 ◁ C4096 ◁ C8192

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16387
transitive degreeμt(G) = 49152

A minimal faithful permutation representation — 2 generators on 16387 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220,1221,1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,12136,8040,16232)(3945,12137,8041,16233)(3946,12138,8042,16234)(3947,12139,8043,16235)(3948,12140,8044,16236)(3949,12141,8045,16237)(3950,12142,8046,16238)(3951,12143,8047,16239)(3952,12144,8048,16240)(3953,12145,8049,16241)(3954,12146,8050,16242)(3955,12147,8051,16243)(3956,12148,8052,16244)(3957,12149,8053,16245)(3958,12150,8054,16246)(3959,12151,8055,16247)(3960,12152,8056,16248)(3961,12153,8057,16249)(3962,12154,8058,16250)(3963,12155,8059,16251)(3964,12156,8060,16252)(3965,12157,8061,16253)(3966,12158,8062,16254)(3967,12159,8063,16255)(3968,12160,8064,16256)(3969,12161,8065,16257)(3970,12162,8066,16258)(3971,12163,8067,16259)(3972,12164,8068,16260)(3973,12165,8069,16261)(3974,12166,8070,16262)(3975,12167,8071,16263)(3976,12168,8072,16264)(3977,12169,8073,16265)(3978,12170,8074,16266)(3979,12171,8075,16267)(3980,12172,8076,16268)(3981,12173,8077,16269)(3982,12174,8078,16270)(3983,12175,8079,16271)(3984,12176,8080,16272)(3985,12177,8081,16273)(3986,12178,8082,16274)(3987,12179,8083,16275)(3988,12180,8084,16276)(3989,12181,8085,16277)(3990,12182,8086,16278)(3991,12183,8087,16279)(3992,12184,8088,16280)(3993,12185,8089,16281)(3994,12186,8090,16282)(3995,12187,8091,16283)(3996,12188,8092,16284)(3997,12189,8093,16285)(3998,12190,8094,16286)(3999,12191,8095,16287)(4000,12192,8096,16288)(4001,12193,8097,16289)(4002,12194,8098,16290)(4003,12195,8099,16291)(4004,12196,8100,16292)(4005,12197,8101,16293)(4006,12198,8102,16294)(4007,12199,8103,16295)(4008,12200,8104,16296)(4009,12201,8105,16297)(4010,12202,8106,16298)(4011,12203,8107,16299)(4012,12204,8108,16300)(4013,12205,8109,16301)(4014,12206,8110,16302)(4015,12207,8111,16303)(4016,12208,8112,16304)(4017,12209,8113,16305)(4018,12210,8114,16306)(4019,12211,8115,16307)(4020,12212,8116,16308)(4021,12213,8117,16309)(4022,12214,8118,16310)(4023,12215,8119,16311)(4024,12216,8120,16312)(4025,12217,8121,16313)(4026,12218,8122,16314)(4027,12219,8123,16315)(4028,12220,8124,16316)(4029,12221,8125,16317)(4030,12222,8126,16318)(4031,12223,8127,16319)(4032,12224,8128,16320)(4033,12225,8129,16321)(4034,12226,8130,16322)(4035,12227,8131,16323)(4036,12228,8132,16324)(4037,12229,8133,16325)(4038,12230,8134,16326)(4039,12231,8135,16327)(4040,12232,8136,16328)(4041,12233,8137,16329)(4042,12234,8138,16330)(4043,12235,8139,16331)(4044,12236,8140,16332)(4045,12237,8141,16333)(4046,12238,8142,16334)(4047,12239,8143,16335)(4048,12240,8144,16336)(4049,12241,8145,16337)(4050,12242,8146,16338)(4051,12243,8147,16339)(4052,12244,8148,16340)(4053,12245,8149,16341)(4054,12246,8150,16342)(4055,12247,8151,16343)(4056,12248,8152,16344)(4057,12249,8153,16345)(4058,12250,8154,16346)(4059,12251,8155,16347)(4060,12252,8156,16348)(4061,12253,8157,16349)(4062,12254,8158,16350)(4063,12255,8159,16351)(4064,12256,8160,16352)(4065,12257,8161,16353)(4066,12258,8162,16354)(4067,12259,8163,16355)(4068,12260,8164,16356)(4069,12261,8165,16357)(4070,12262,8166,16358)(4071,12263,8167,16359)(4072,12264,8168,16360)(4073,12265,8169,16361)(4074,12266,8170,16362)(4075,12267,8171,16363)(4076,12268,8172,16364)(4077,12269,8173,16365)(4078,12270,8174,16366)(4079,12271,8175,16367)(4080,12272,8176,16368)(4081,12273,8177,16369)(4082,12274,8178,16370)(4083,12275,8179,16371)(4084,12276,8180,16372)(4085,12277,8181,16373)(4086,12278,8182,16374)(4087,12279,8183,16375)(4088,12280,8184,16376)(4089,12281,8185,16377)(4090,12282,8186,16378)(4091,12283,8187,16379)(4092,12284,8188,16380)(4093,12285,8189,16381)(4094,12286,8190,16382)(4095,12287,8191,16383)(4096,12288,8192,16384)(16386,16387)

Polycyclic presentation:

2^14,3|a^2=n,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=jk,j^2=kl,k^2=lm,l^2=mn,m^2=n,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,i^a=ij,j^a=jk,k^a=kl,l^a=lm,m^a=mn,o^a=o^2

Conjugacy classes & character table

12291 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m", "n", "o");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.14);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6*F.7);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetPower(coll, 7, F.8*F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11*F.12);;
gap> SetPower(coll, 11, F.12*F.13);;
gap> SetPower(coll, 12, F.13*F.14);;
gap> SetPower(coll, 13, F.14);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 11, 1, F.11*F.12);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> SetConjugate(coll, 13, 1, F.13*F.14);;
gap> SetConjugate(coll, 15, 1, F.15^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);