Dic₁₅₃₈₄ #72293435

order	61536
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₃₈₄

Splittings

Hall	C₃×C₆₄₁ ⋊ Q₃₂
Sylow	C₆₄₁ ⋊ Dic₂₄
Sylow	C₃ ⋊ Dic₅₁₂₈

Subgroups

	subgroup	G/H
center	C₂	D₁₅₃₈₄
commutator, cosocle	C₃×C₈×C₆₄₁	C₂²
Frattini	C₈	C₂×D₁₉₂₃
Fitting	C₃×C₁₆×C₆₄₁	C₂
socle, prime radical	C₂×C₃×C₆₄₁	D₈
hypercenter	C₁₆	D₁₉₂₃
hypocenter	C₃×C₆₄₁	Q₃₂
LCS	G ▷ C₃×C₈×C₆₄₁ ▷ C₃×C₄×C₆₄₁ ▷ C₂×C₃×C₆₄₁ ▷ C₃×C₆₄₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 676
transitive degree	μ_t(G) = 61536

A minimal faithful permutation representation — 2 generators on 676 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18)(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);(1,17,9,25)(2,18,10,26)(3,19,11,27)(4,20,12,28)(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(34,35)(37,676)(38,675)(39,674)(40,673)(41,672)(42,671)(43,670)(44,669)(45,668)(46,667)(47,666)(48,665)(49,664)(50,663)(51,662)(52,661)(53,660)(54,659)(55,658)(56,657)(57,656)(58,655)(59,654)(60,653)(61,652)(62,651)(63,650)(64,649)(65,648)(66,647)(67,646)(68,645)(69,644)(70,643)(71,642)(72,641)(73,640)(74,639)(75,638)(76,637)(77,636)(78,635)(79,634)(80,633)(81,632)(82,631)(83,630)(84,629)(85,628)(86,627)(87,626)(88,625)(89,624)(90,623)(91,622)(92,621)(93,620)(94,619)(95,618)(96,617)(97,616)(98,615)(99,614)(100,613)(101,612)(102,611)(103,610)(104,609)(105,608)(106,607)(107,606)(108,605)(109,604)(110,603)(111,602)(112,601)(113,600)(114,599)(115,598)(116,597)(117,596)(118,595)(119,594)(120,593)(121,592)(122,591)(123,590)(124,589)(125,588)(126,587)(127,586)(128,585)(129,584)(130,583)(131,582)(132,581)(133,580)(134,579)(135,578)(136,577)(137,576)(138,575)(139,574)(140,573)(141,572)(142,571)(143,570)(144,569)(145,568)(146,567)(147,566)(148,565)(149,564)(150,563)(151,562)(152,561)(153,560)(154,559)(155,558)(156,557)(157,556)(158,555)(159,554)(160,553)(161,552)(162,551)(163,550)(164,549)(165,548)(166,547)(167,546)(168,545)(169,544)(170,543)(171,542)(172,541)(173,540)(174,539)(175,538)(176,537)(177,536)(178,535)(179,534)(180,533)(181,532)(182,531)(183,530)(184,529)(185,528)(186,527)(187,526)(188,525)(189,524)(190,523)(191,522)(192,521)(193,520)(194,519)(195,518)(196,517)(197,516)(198,515)(199,514)(200,513)(201,512)(202,511)(203,510)(204,509)(205,508)(206,507)(207,506)(208,505)(209,504)(210,503)(211,502)(212,501)(213,500)(214,499)(215,498)(216,497)(217,496)(218,495)(219,494)(220,493)(221,492)(222,491)(223,490)(224,489)(225,488)(226,487)(227,486)(228,485)(229,484)(230,483)(231,482)(232,481)(233,480)(234,479)(235,478)(236,477)(237,476)(238,475)(239,474)(240,473)(241,472)(242,471)(243,470)(244,469)(245,468)(246,467)(247,466)(248,465)(249,464)(250,463)(251,462)(252,461)(253,460)(254,459)(255,458)(256,457)(257,456)(258,455)(259,454)(260,453)(261,452)(262,451)(263,450)(264,449)(265,448)(266,447)(267,446)(268,445)(269,444)(270,443)(271,442)(272,441)(273,440)(274,439)(275,438)(276,437)(277,436)(278,435)(279,434)(280,433)(281,432)(282,431)(283,430)(284,429)(285,428)(286,427)(287,426)(288,425)(289,424)(290,423)(291,422)(292,421)(293,420)(294,419)(295,418)(296,417)(297,416)(298,415)(299,414)(300,413)(301,412)(302,411)(303,410)(304,409)(305,408)(306,407)(307,406)(308,405)(309,404)(310,403)(311,402)(312,401)(313,400)(314,399)(315,398)(316,397)(317,396)(318,395)(319,394)(320,393)(321,392)(322,391)(323,390)(324,389)(325,388)(326,387)(327,386)(328,385)(329,384)(330,383)(331,382)(332,381)(333,380)(334,379)(335,378)(336,377)(337,376)(338,375)(339,374)(340,373)(341,372)(342,371)(343,370)(344,369)(345,368)(346,367)(347,366)(348,365)(349,364)(350,363)(351,362)(352,361)(353,360)(354,359)(355,358)(356,357)

Polycyclic presentation:

2^5,3,641|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^640

Conjugacy classes & character table

15387 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 641]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^640);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15384 #72293435