Dic₃₈₅₈ #72215134

order	15432
automorphism	Aut(G) order 19814688
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₃₈₅₈

Splittings

Hall	C₃×C₆₄₃ ⋊ Q₈
Sylow	C₆₄₃ ⋊ Dic₆
Sylow	C₃ ⋊ Dic₁₂₈₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₉₂₉
commutator, socle, prime radical, cosocle	C₂×C₃×C₆₄₃	C₂²
Fitting	C₃×C₄×C₆₄₃	C₂
hypercenter	C₄	D₁₉₂₉
hypocenter	C₃×C₆₄₃	Q₈
LCS	G ▷ C₂×C₃×C₆₄₃ ▷ C₃×C₆₄₃
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 654
transitive degree	μ_t(G) = 15432

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

Polycyclic presentation:

2^3,3,643|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^642

Conjugacy classes & character table

3861 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₇₁₆, Dic₁₁₅₇₄
maximal normal subgroup of	Dic₇₇₁₆
maximal quotient of	Dic₁₁₅₇₄

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, 643]);;
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.5^642);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic3858 #72215134