Dic₈₆₂₃ #46828461

order	34492
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₆₂₃
unambiguous semidirect product	C₈₆₂₃⋊C₄

Splittings

derived, Sylow

C₈₆₂₃ ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₆₂₃
commutator, hypocenter	C₈₆₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₈₆₂₃	C₂
LCS	G ▷ C₈₆₂₃
UCS	C₁ ◁ C₂

6 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8627
transitive degree	μ_t(G) = 34492

A minimal faithful permutation representation — 2 generators on 8627 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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623)(4011,4622)(4012,4621)(4013,4620)(4014,4619)(4015,4618)(4016,4617)(4017,4616)(4018,4615)(4019,4614)(4020,4613)(4021,4612)(4022,4611)(4023,4610)(4024,4609)(4025,4608)(4026,4607)(4027,4606)(4028,4605)(4029,4604)(4030,4603)(4031,4602)(4032,4601)(4033,4600)(4034,4599)(4035,4598)(4036,4597)(4037,4596)(4038,4595)(4039,4594)(4040,4593)(4041,4592)(4042,4591)(4043,4590)(4044,4589)(4045,4588)(4046,4587)(4047,4586)(4048,4585)(4049,4584)(4050,4583)(4051,4582)(4052,4581)(4053,4580)(4054,4579)(4055,4578)(4056,4577)(4057,4576)(4058,4575)(4059,4574)(4060,4573)(4061,4572)(4062,4571)(4063,4570)(4064,4569)(4065,4568)(4066,4567)(4067,4566)(4068,4565)(4069,4564)(4070,4563)(4071,4562)(4072,4561)(4073,4560)(4074,4559)(4075,4558)(4076,4557)(4077,4556)(4078,4555)(4079,4554)(4080,4553)(4081,4552)(4082,4551)(4083,4550)(4084,4549)(4085,4548)(4086,4547)(4087,4546)(4088,4545)(4089,4544)(4090,4543)(4091,4542)(4092,4541)(4093,4540)(4094,4539)(4095,4538)(4096,4537)(4097,4536)(4098,4535)(4099,4534)(4100,4533)(4101,4532)(4102,4531)(4103,4530)(4104,4529)(4105,4528)(4106,4527)(4107,4526)(4108,4525)(4109,4524)(4110,4523)(4111,4522)(4112,4521)(4113,4520)(4114,4519)(4115,4518)(4116,4517)(4117,4516)(4118,4515)(4119,4514)(4120,4513)(4121,4512)(4122,4511)(4123,4510)(4124,4509)(4125,4508)(4126,4507)(4127,4506)(4128,4505)(4129,4504)(4130,4503)(4131,4502)(4132,4501)(4133,4500)(4134,4499)(4135,4498)(4136,4497)(4137,4496)(4138,4495)(4139,4494)(4140,4493)(4141,4492)(4142,4491)(4143,4490)(4144,4489)(4145,4488)(4146,4487)(4147,4486)(4148,4485)(4149,4484)(4150,4483)(4151,4482)(4152,4481)(4153,4480)(4154,4479)(4155,4478)(4156,4477)(4157,4476)(4158,4475)(4159,4474)(4160,4473)(4161,4472)(4162,4471)(4163,4470)(4164,4469)(4165,4468)(4166,4467)(4167,4466)(4168,4465)(4169,4464)(4170,4463)(4171,4462)(4172,4461)(4173,4460)(4174,4459)(4175,4458)(4176,4457)(4177,4456)(4178,4455)(4179,4454)(4180,4453)(4181,4452)(4182,4451)(4183,4450)(4184,4449)(4185,4448)(4186,4447)(4187,4446)(4188,4445)(4189,4444)(4190,4443)(4191,4442)(4192,4441)(4193,4440)(4194,4439)(4195,4438)(4196,4437)(4197,4436)(4198,4435)(4199,4434)(4200,4433)(4201,4432)(4202,4431)(4203,4430)(4204,4429)(4205,4428)(4206,4427)(4207,4426)(4208,4425)(4209,4424)(4210,4423)(4211,4422)(4212,4421)(4213,4420)(4214,4419)(4215,4418)(4216,4417)(4217,4416)(4218,4415)(4219,4414)(4220,4413)(4221,4412)(4222,4411)(4223,4410)(4224,4409)(4225,4408)(4226,4407)(4227,4406)(4228,4405)(4229,4404)(4230,4403)(4231,4402)(4232,4401)(4233,4400)(4234,4399)(4235,4398)(4236,4397)(4237,4396)(4238,4395)(4239,4394)(4240,4393)(4241,4392)(4242,4391)(4243,4390)(4244,4389)(4245,4388)(4246,4387)(4247,4386)(4248,4385)(4249,4384)(4250,4383)(4251,4382)(4252,4381)(4253,4380)(4254,4379)(4255,4378)(4256,4377)(4257,4376)(4258,4375)(4259,4374)(4260,4373)(4261,4372)(4262,4371)(4263,4370)(4264,4369)(4265,4368)(4266,4367)(4267,4366)(4268,4365)(4269,4364)(4270,4363)(4271,4362)(4272,4361)(4273,4360)(4274,4359)(4275,4358)(4276,4357)(4277,4356)(4278,4355)(4279,4354)(4280,4353)(4281,4352)(4282,4351)(4283,4350)(4284,4349)(4285,4348)(4286,4347)(4287,4346)(4288,4345)(4289,4344)(4290,4343)(4291,4342)(4292,4341)(4293,4340)(4294,4339)(4295,4338)(4296,4337)(4297,4336)(4298,4335)(4299,4334)(4300,4333)(4301,4332)(4302,4331)(4303,4330)(4304,4329)(4305,4328)(4306,4327)(4307,4326)(4308,4325)(4309,4324)(4310,4323)(4311,4322)(4312,4321)(4313,4320)(4314,4319)(4315,4318)(4316,4317)

Polycyclic presentation:

2^2,8623|a^2=b,c^a=c^8622

Conjugacy classes & character table

8626 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34492, 1);

Dic8623 #46828461