Dic₁₄₈₀₈ #72292015

order	59232
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₈₀₈

Splittings

Hall	C₃×C₆₁₇ ⋊ Q₃₂
Sylow	C₆₁₇ ⋊ Dic₂₄
Sylow	C₃ ⋊ Dic₄₉₃₆

Subgroups

	subgroup	G/H
center	C₂	D₁₄₈₀₈
commutator, cosocle	C₃×C₈×C₆₁₇	C₂²
Frattini	C₈	C₂×D₁₈₅₁
Fitting	C₃×C₁₆×C₆₁₇	C₂
socle, prime radical	C₂×C₃×C₆₁₇	D₈
hypercenter	C₁₆	D₁₈₅₁
hypocenter	C₃×C₆₁₇	Q₃₂
LCS	G ▷ C₃×C₈×C₆₁₇ ▷ C₃×C₄×C₆₁₇ ▷ C₂×C₃×C₆₁₇ ▷ C₃×C₆₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 652
transitive degree	μ_t(G) = 59232

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

Polycyclic presentation:

2^5,3,617|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^616

Conjugacy classes & character table

14811 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, 617]);;
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.7^616);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14808 #72292015