Dic₄₃₉₈ #72215849

order	17592
automorphism	Aut(G) order 25754688
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₄₃₉₈

Splittings

Hall	C₃×C₇₃₃ ⋊ Q₈
Sylow	C₇₃₃ ⋊ Dic₆
Sylow	C₃ ⋊ Dic₁₄₆₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₁₉₉
commutator, socle, prime radical, cosocle	C₂×C₃×C₇₃₃	C₂²
Fitting	C₃×C₄×C₇₃₃	C₂
hypercenter	C₄	D₂₁₉₉
hypocenter	C₃×C₇₃₃	Q₈
LCS	G ▷ C₂×C₃×C₇₃₃ ▷ C₃×C₇₃₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 744
transitive degree	μ_t(G) = 17592

A minimal faithful permutation representation — 2 generators on 744 points:

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,11)(13,744)(14,743)(15,742)(16,741)(17,740)(18,739)(19,738)(20,737)(21,736)(22,735)(23,734)(24,733)(25,732)(26,731)(27,730)(28,729)(29,728)(30,727)(31,726)(32,725)(33,724)(34,723)(35,722)(36,721)(37,720)(38,719)(39,718)(40,717)(41,716)(42,715)(43,714)(44,713)(45,712)(46,711)(47,710)(48,709)(49,708)(50,707)(51,706)(52,705)(53,704)(54,703)(55,702)(56,701)(57,700)(58,699)(59,698)(60,697)(61,696)(62,695)(63,694)(64,693)(65,692)(66,691)(67,690)(68,689)(69,688)(70,687)(71,686)(72,685)(73,684)(74,683)(75,682)(76,681)(77,680)(78,679)(79,678)(80,677)(81,676)(82,675)(83,674)(84,673)(85,672)(86,671)(87,670)(88,669)(89,668)(90,667)(91,666)(92,665)(93,664)(94,663)(95,662)(96,661)(97,660)(98,659)(99,658)(100,657)(101,656)(102,655)(103,654)(104,653)(105,652)(106,651)(107,650)(108,649)(109,648)(110,647)(111,646)(112,645)(113,644)(114,643)(115,642)(116,641)(117,640)(118,639)(119,638)(120,637)(121,636)(122,635)(123,634)(124,633)(125,632)(126,631)(127,630)(128,629)(129,628)(130,627)(131,626)(132,625)(133,624)(134,623)(135,622)(136,621)(137,620)(138,619)(139,618)(140,617)(141,616)(142,615)(143,614)(144,613)(145,612)(146,611)(147,610)(148,609)(149,608)(150,607)(151,606)(152,605)(153,604)(154,603)(155,602)(156,601)(157,600)(158,599)(159,598)(160,597)(161,596)(162,595)(163,594)(164,593)(165,592)(166,591)(167,590)(168,589)(169,588)(170,587)(171,586)(172,585)(173,584)(174,583)(175,582)(176,581)(177,580)(178,579)(179,578)(180,577)(181,576)(182,575)(183,574)(184,573)(185,572)(186,571)(187,570)(188,569)(189,568)(190,567)(191,566)(192,565)(193,564)(194,563)(195,562)(196,561)(197,560)(198,559)(199,558)(200,557)(201,556)(202,555)(203,554)(204,553)(205,552)(206,551)(207,550)(208,549)(209,548)(210,547)(211,546)(212,545)(213,544)(214,543)(215,542)(216,541)(217,540)(218,539)(219,538)(220,537)(221,536)(222,535)(223,534)(224,533)(225,532)(226,531)(227,530)(228,529)(229,528)(230,527)(231,526)(232,525)(233,524)(234,523)(235,522)(236,521)(237,520)(238,519)(239,518)(240,517)(241,516)(242,515)(243,514)(244,513)(245,512)(246,511)(247,510)(248,509)(249,508)(250,507)(251,506)(252,505)(253,504)(254,503)(255,502)(256,501)(257,500)(258,499)(259,498)(260,497)(261,496)(262,495)(263,494)(264,493)(265,492)(266,491)(267,490)(268,489)(269,488)(270,487)(271,486)(272,485)(273,484)(274,483)(275,482)(276,481)(277,480)(278,479)(279,478)(280,477)(281,476)(282,475)(283,474)(284,473)(285,472)(286,471)(287,470)(288,469)(289,468)(290,467)(291,466)(292,465)(293,464)(294,463)(295,462)(296,461)(297,460)(298,459)(299,458)(300,457)(301,456)(302,455)(303,454)(304,453)(305,452)(306,451)(307,450)(308,449)(309,448)(310,447)(311,446)(312,445)(313,444)(314,443)(315,442)(316,441)(317,440)(318,439)(319,438)(320,437)(321,436)(322,435)(323,434)(324,433)(325,432)(326,431)(327,430)(328,429)(329,428)(330,427)(331,426)(332,425)(333,424)(334,423)(335,422)(336,421)(337,420)(338,419)(339,418)(340,417)(341,416)(342,415)(343,414)(344,413)(345,412)(346,411)(347,410)(348,409)(349,408)(350,407)(351,406)(352,405)(353,404)(354,403)(355,402)(356,401)(357,400)(358,399)(359,398)(360,397)(361,396)(362,395)(363,394)(364,393)(365,392)(366,391)(367,390)(368,389)(369,388)(370,387)(371,386)(372,385)(373,384)(374,383)(375,382)(376,381)(377,380)(378,379)

Polycyclic presentation:

2^3,3,733|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^732

Conjugacy classes & character table

4401 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₇₉₆, Dic₁₃₁₉₄
maximal normal subgroup of	Dic₈₇₉₆
maximal quotient of	Dic₁₃₁₉₄

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 733]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^732);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic4398 #72215849