Dic₄₃₅₂ #72215832

order	17408
automorphism	Aut(G) order 35651584
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₅₂

Splittings

Sylow

C₁₇ ⋊ Q₁₀₂₄

Subgroups

	subgroup	G/H
center	C₂	D₄₃₅₂
commutator, cosocle	C₁₇×C₂₅₆	C₂²
Frattini	C₂₅₆	C₂×D₁₇
Fitting	C₁₇×C₅₁₂	C₂
socle, prime radical	C₂×C₁₇	D₂₅₆
hypercenter	C₅₁₂	D₁₇
hypocenter	C₁₇	Q₁₀₂₄
LCS	G ▷ C₁₇×C₂₅₆ ▷ C₁₇×C₁₂₈ ▷ C₁₇×C₆₄ ▷ C₁₇×C₃₂ ▷ C₁₆×C₁₇ ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ C₆₄ ◁ C₁₂₈ ◁ C₂₅₆ ◁ C₅₁₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1041
transitive degree	μ_t(G) = 17408

A minimal faithful permutation representation — 2 generators on 1041 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)(513,1024,1023,1022,1021,1020,1019,1018,1017,1016,1015,1014,1013,1012,1011,1010,1009,1008,1007,1006,1005,1004,1003,1002,1001,1000,999,998,997,996,995,994,993,992,991,990,989,988,987,986,985,984,983,982,981,980,979,978,977,976,975,974,973,972,971,970,969,968,967,966,965,964,963,962,961,960,959,958,957,956,955,954,953,952,951,950,949,948,947,946,945,944,943,942,941,940,939,938,937,936,935,934,933,932,931,930,929,928,927,926,925,924,923,922,921,920,919,918,917,916,915,914,913,912,911,910,909,908,907,906,905,904,903,902,901,900,899,898,897,896,895,894,893,892,891,890,889,888,887,886,885,884,883,882,881,880,879,878,877,876,875,874,873,872,871,870,869,868,867,866,865,864,863,862,861,860,859,858,857,856,855,854,853,852,851,850,849,848,847,846,845,844,843,842,841,840,839,838,837,836,835,834,833,832,831,830,829,828,827,826,825,824,823,822,821,820,819,818,817,816,815,814,813,812,811,810,809,808,807,806,805,804,803,802,801,800,799,798,797,796,795,794,793,792,791,790,789,788,787,786,785,784,783,782,781,780,779,778,777,776,775,774,773,772,771,770,769,768,767,766,765,764,763,762,761,760,759,758,757,756,755,754,753,752,751,750,749,748,747,746,745,744,743,742,741,740,739,738,737,736,735,734,733,732,731,730,729,728,727,726,725,724,723,722,721,720,719,718,717,716,715,714,713,712,711,710,709,708,707,706,705,704,703,702,701,700,699,698,697,696,695,694,693,692,691,690,689,688,687,686,685,684,683,682,681,680,679,678,677,676,675,674,673,672,671,670,669,668,667,666,665,664,663,662,661,660,659,658,657,656,655,654,653,652,651,650,649,648,647,646,645,644,643,642,641,640,639,638,637,636,635,634,633,632,631,630,629,628,627,626,625,624,623,622,621,620,619,618,617,616,615,614,613,612,611,610,609,608,607,606,605,604,603,602,601,600,599,598,597,596,595,594,593,592,591,590,589,588,587,586,585,584,583,582,581,580,579,578,577,576,575,574,573,572,571,570,569,568,567,566,565,564,563,562,561,560,559,558,557,556,555,554,553,552,551,550,549,548,547,546,545,544,543,542,541,540,539,538,537,536,535,534,533,532,531,530,529,528,527,526,525,524,523,522,521,520,519,518,517,516,515,514)(1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041);(1,513,257,769)(2,514,258,770)(3,515,259,771)(4,516,260,772)(5,517,261,773)(6,518,262,774)(7,519,263,775)(8,520,264,776)(9,521,265,777)(10,522,266,778)(11,523,267,779)(12,524,268,780)(13,525,269,781)(14,526,270,782)(15,527,271,783)(16,528,272,784)(17,529,273,785)(18,530,274,786)(19,531,275,787)(20,532,276,788)(21,533,277,789)(22,534,278,790)(23,535,279,791)(24,536,280,792)(25,537,281,793)(26,538,282,794)(27,539,283,795)(28,540,284,796)(29,541,285,797)(30,542,286,798)(31,543,287,799)(32,544,288,800)(33,545,289,801)(34,546,290,802)(35,547,291,803)(36,548,292,804)(37,549,293,805)(38,550,294,806)(39,551,295,807)(40,552,296,808)(41,553,297,809)(42,554,298,810)(43,555,299,811)(44,556,300,812)(45,557,301,813)(46,558,302,814)(47,559,303,815)(48,560,304,816)(49,561,305,817)(50,562,306,818)(51,563,307,819)(52,564,308,820)(53,565,309,821)(54,566,310,822)(55,567,311,823)(56,568,312,824)(57,569,313,825)(58,570,314,826)(59,571,315,827)(60,572,316,828)(61,573,317,829)(62,574,318,830)(63,575,319,831)(64,576,320,832)(65,577,321,833)(66,578,322,834)(67,579,323,835)(68,580,324,836)(69,581,325,837)(70,582,326,838)(71,583,327,839)(72,584,328,840)(73,585,329,841)(74,586,330,842)(75,587,331,843)(76,588,332,844)(77,589,333,845)(78,590,334,846)(79,591,335,847)(80,592,336,848)(81,593,337,849)(82,594,338,850)(83,595,339,851)(84,596,340,852)(85,597,341,853)(86,598,342,854)(87,599,343,855)(88,600,344,856)(89,601,345,857)(90,602,346,858)(91,603,347,859)(92,604,348,860)(93,605,349,861)(94,606,350,862)(95,607,351,863)(96,608,352,864)(97,609,353,865)(98,610,354,866)(99,611,355,867)(100,612,356,868)(101,613,357,869)(102,614,358,870)(103,615,359,871)(104,616,360,872)(105,617,361,873)(106,618,362,874)(107,619,363,875)(108,620,364,876)(109,621,365,877)(110,622,366,878)(111,623,367,879)(112,624,368,880)(113,625,369,881)(114,626,370,882)(115,627,371,883)(116,628,372,884)(117,629,373,885)(118,630,374,886)(119,631,375,887)(120,632,376,888)(121,633,377,889)(122,634,378,890)(123,635,379,891)(124,636,380,892)(125,637,381,893)(126,638,382,894)(127,639,383,895)(128,640,384,896)(129,641,385,897)(130,642,386,898)(131,643,387,899)(132,644,388,900)(133,645,389,901)(134,646,390,902)(135,647,391,903)(136,648,392,904)(137,649,393,905)(138,650,394,906)(139,651,395,907)(140,652,396,908)(141,653,397,909)(142,654,398,910)(143,655,399,911)(144,656,400,912)(145,657,401,913)(146,658,402,914)(147,659,403,915)(148,660,404,916)(149,661,405,917)(150,662,406,918)(151,663,407,919)(152,664,408,920)(153,665,409,921)(154,666,410,922)(155,667,411,923)(156,668,412,924)(157,669,413,925)(158,670,414,926)(159,671,415,927)(160,672,416,928)(161,673,417,929)(162,674,418,930)(163,675,419,931)(164,676,420,932)(165,677,421,933)(166,678,422,934)(167,679,423,935)(168,680,424,936)(169,681,425,937)(170,682,426,938)(171,683,427,939)(172,684,428,940)(173,685,429,941)(174,686,430,942)(175,687,431,943)(176,688,432,944)(177,689,433,945)(178,690,434,946)(179,691,435,947)(180,692,436,948)(181,693,437,949)(182,694,438,950)(183,695,439,951)(184,696,440,952)(185,697,441,953)(186,698,442,954)(187,699,443,955)(188,700,444,956)(189,701,445,957)(190,702,446,958)(191,703,447,959)(192,704,448,960)(193,705,449,961)(194,706,450,962)(195,707,451,963)(196,708,452,964)(197,709,453,965)(198,710,454,966)(199,711,455,967)(200,712,456,968)(201,713,457,969)(202,714,458,970)(203,715,459,971)(204,716,460,972)(205,717,461,973)(206,718,462,974)(207,719,463,975)(208,720,464,976)(209,721,465,977)(210,722,466,978)(211,723,467,979)(212,724,468,980)(213,725,469,981)(214,726,470,982)(215,727,471,983)(216,728,472,984)(217,729,473,985)(218,730,474,986)(219,731,475,987)(220,732,476,988)(221,733,477,989)(222,734,478,990)(223,735,479,991)(224,736,480,992)(225,737,481,993)(226,738,482,994)(227,739,483,995)(228,740,484,996)(229,741,485,997)(230,742,486,998)(231,743,487,999)(232,744,488,1000)(233,745,489,1001)(234,746,490,1002)(235,747,491,1003)(236,748,492,1004)(237,749,493,1005)(238,750,494,1006)(239,751,495,1007)(240,752,496,1008)(241,753,497,1009)(242,754,498,1010)(243,755,499,1011)(244,756,500,1012)(245,757,501,1013)(246,758,502,1014)(247,759,503,1015)(248,760,504,1016)(249,761,505,1017)(250,762,506,1018)(251,763,507,1019)(252,764,508,1020)(253,765,509,1021)(254,766,510,1022)(255,767,511,1023)(256,768,512,1024)(1026,1041)(1027,1040)(1028,1039)(1029,1038)(1030,1037)(1031,1036)(1032,1035)(1033,1034)

Polycyclic presentation:

2^10,17|a^2=j,b^2=cd,c^2=de,d^2=ef,e^2=fg,f^2=gh,g^2=hi,h^2=ij,i^2=j,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,g^a=gh,h^a=hi,i^a=ij,k^a=k^16

Conjugacy classes & character table

4355 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₀₄, Dic₁₃₀₅₆
maximal normal subgroup of	Dic₈₇₀₄
maximal quotient of	Dic₁₃₀₅₆

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 17]);;
gap> SetPower(coll, 1, F.10);;
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);;
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, 11, 1, F.11^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4352 #72215832