Q₃₂₇₆₈ #72210177

order	32768
nilpotency class	14
automorphism	Aut(G) order 134217728
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

quaternion	Q₃₂₇₆₈
dicyclic	Dic₈₁₉₂

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂	D₈₁₉₂
commutator, Frattini, cosocle	C₈₁₉₂	C₂²
LCS	G ▷ C₈₁₉₂ ▷ C₄₀₉₆ ▷ C₂₀₄₈ ▷ C₁₀₂₄ ▷ C₅₁₂ ▷ C₂₅₆ ▷ C₁₂₈ ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈ ◁ C₂₅₆ ◁ C₅₁₂ ◁ C₁₀₂₄ ◁ C₂₀₄₈ ◁ C₄₀₉₆ ◁ C₈₁₉₂ ◁ G
LECS	G ▷ C₈₁₉₂ ▷ C₄₀₉₆ ▷ C₂₀₄₈ ▷ C₁₀₂₄ ▷ C₅₁₂ ▷ C₂₅₆ ▷ C₁₂₈ ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32768
transitive degree	μ_t(G) = 32768

A minimal faithful permutation representation — 2 generators on 32768 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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8018,24402,16210,32594)(8019,24403,16211,32595)(8020,24404,16212,32596)(8021,24405,16213,32597)(8022,24406,16214,32598)(8023,24407,16215,32599)(8024,24408,16216,32600)(8025,24409,16217,32601)(8026,24410,16218,32602)(8027,24411,16219,32603)(8028,24412,16220,32604)(8029,24413,16221,32605)(8030,24414,16222,32606)(8031,24415,16223,32607)(8032,24416,16224,32608)(8033,24417,16225,32609)(8034,24418,16226,32610)(8035,24419,16227,32611)(8036,24420,16228,32612)(8037,24421,16229,32613)(8038,24422,16230,32614)(8039,24423,16231,32615)(8040,24424,16232,32616)(8041,24425,16233,32617)(8042,24426,16234,32618)(8043,24427,16235,32619)(8044,24428,16236,32620)(8045,24429,16237,32621)(8046,24430,16238,32622)(8047,24431,16239,32623)(8048,24432,16240,32624)(8049,24433,16241,32625)(8050,24434,16242,32626)(8051,24435,16243,32627)(8052,24436,16244,32628)(8053,24437,16245,32629)(8054,24438,16246,32630)(8055,24439,16247,32631)(8056,24440,16248,32632)(8057,24441,16249,32633)(8058,24442,16250,32634)(8059,24443,16251,32635)(8060,24444,16252,32636)(8061,24445,16253,32637)(8062,24446,16254,32638)(8063,24447,16255,32639)(8064,24448,16256,32640)(8065,24449,16257,32641)(8066,24450,16258,32642)(8067,24451,16259,32643)(8068,24452,16260,32644)(8069,24453,16261,32645)(8070,24454,16262,32646)(8071,24455,16263,32647)(8072,24456,16264,32648)(8073,24457,16265,32649)(8074,24458,16266,32650)(8075,24459,16267,32651)(8076,24460,16268,32652)(8077,24461,16269,32653)(8078,24462,16270,32654)(8079,24463,16271,32655)(8080,24464,16272,32656)(8081,24465,16273,32657)(8082,24466,16274,32658)(8083,24467,16275,32659)(8084,24468,16276,32660)(8085,24469,16277,32661)(8086,24470,16278,32662)(8087,24471,16279,32663)(8088,24472,16280,32664)(8089,24473,16281,32665)(8090,24474,16282,32666)(8091,24475,16283,32667)(8092,24476,16284,32668)(8093,24477,16285,32669)(8094,24478,16286,32670)(8095,24479,16287,32671)(8096,24480,16288,32672)(8097,24481,16289,32673)(8098,24482,16290,32674)(8099,24483,16291,32675)(8100,24484,16292,32676)(8101,24485,16293,32677)(8102,24486,16294,32678)(8103,24487,16295,32679)(8104,24488,16296,32680)(8105,24489,16297,32681)(8106,24490,16298,32682)(8107,24491,16299,32683)(8108,24492,16300,32684)(8109,24493,16301,32685)(8110,24494,16302,32686)(8111,24495,16303,32687)(8112,24496,16304,32688)(8113,24497,16305,32689)(8114,24498,16306,32690)(8115,24499,16307,32691)(8116,24500,16308,32692)(8117,24501,16309,32693)(8118,24502,16310,32694)(8119,24503,16311,32695)(8120,24504,16312,32696)(8121,24505,16313,32697)(8122,24506,16314,32698)(8123,24507,16315,32699)(8124,24508,16316,32700)(8125,24509,16317,32701)(8126,24510,16318,32702)(8127,24511,16319,32703)(8128,24512,16320,32704)(8129,24513,16321,32705)(8130,24514,16322,32706)(8131,24515,16323,32707)(8132,24516,16324,32708)(8133,24517,16325,32709)(8134,24518,16326,32710)(8135,24519,16327,32711)(8136,24520,16328,32712)(8137,24521,16329,32713)(8138,24522,16330,32714)(8139,24523,16331,32715)(8140,24524,16332,32716)(8141,24525,16333,32717)(8142,24526,16334,32718)(8143,24527,16335,32719)(8144,24528,16336,32720)(8145,24529,16337,32721)(8146,24530,16338,32722)(8147,24531,16339,32723)(8148,24532,16340,32724)(8149,24533,16341,32725)(8150,24534,16342,32726)(8151,24535,16343,32727)(8152,24536,16344,32728)(8153,24537,16345,32729)(8154,24538,16346,32730)(8155,24539,16347,32731)(8156,24540,16348,32732)(8157,24541,16349,32733)(8158,24542,16350,32734)(8159,24543,16351,32735)(8160,24544,16352,32736)(8161,24545,16353,32737)(8162,24546,16354,32738)(8163,24547,16355,32739)(8164,24548,16356,32740)(8165,24549,16357,32741)(8166,24550,16358,32742)(8167,24551,16359,32743)(8168,24552,16360,32744)(8169,24553,16361,32745)(8170,24554,16362,32746)(8171,24555,16363,32747)(8172,24556,16364,32748)(8173,24557,16365,32749)(8174,24558,16366,32750)(8175,24559,16367,32751)(8176,24560,16368,32752)(8177,24561,16369,32753)(8178,24562,16370,32754)(8179,24563,16371,32755)(8180,24564,16372,32756)(8181,24565,16373,32757)(8182,24566,16374,32758)(8183,24567,16375,32759)(8184,24568,16376,32760)(8185,24569,16377,32761)(8186,24570,16378,32762)(8187,24571,16379,32763)(8188,24572,16380,32764)(8189,24573,16381,32765)(8190,24574,16382,32766)(8191,24575,16383,32767)(8192,24576,16384,32768)

Polycyclic presentation:

2^15|a^2=o,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=no,n^2=o,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,n^a=no

Conjugacy classes & character table

8195 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, 2]);;
gap> SetPower(coll, 1, F.15);;
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*F.15);;
gap> SetPower(coll, 14, F.15);;
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, 14, 1, F.14*F.15);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Q32768 #72210177

Aliases

Subgroups

Presentation

Conjugacy classes & character table

GAP

Q₃₂₇₆₈ #72210177