Dic₈₃₅₅ #45689205

order	33420
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₃₅₅
unambiguous semidirect product	C₃⋊Dic₂₇₈₅

Splittings

derived, Hall	C₃×C₅×C₅₅₇ ⋊ C₄
Hall	C₅×C₅₅₇ ⋊ Dic₃
Hall	C₃×C₅₅₇ ⋊ Dic₅
Sylow	C₅₅₇ ⋊ Dic₁₅
Hall	C₃×C₅ ⋊ Dic₅₅₇
Sylow	C₅ ⋊ Dic₁₆₇₁
Sylow	C₃ ⋊ Dic₂₇₈₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₃₅₅
commutator, hypocenter	C₃×C₅×C₅₅₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₅₅₇	C₂
LCS	G ▷ C₃×C₅×C₅₅₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 569
transitive degree	μ_t(G) = 33420

A minimal faithful permutation representation — 2 generators on 569 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569);(1,3,2,4)(6,7)(9,12)(10,11)(14,569)(15,568)(16,567)(17,566)(18,565)(19,564)(20,563)(21,562)(22,561)(23,560)(24,559)(25,558)(26,557)(27,556)(28,555)(29,554)(30,553)(31,552)(32,551)(33,550)(34,549)(35,548)(36,547)(37,546)(38,545)(39,544)(40,543)(41,542)(42,541)(43,540)(44,539)(45,538)(46,537)(47,536)(48,535)(49,534)(50,533)(51,532)(52,531)(53,530)(54,529)(55,528)(56,527)(57,526)(58,525)(59,524)(60,523)(61,522)(62,521)(63,520)(64,519)(65,518)(66,517)(67,516)(68,515)(69,514)(70,513)(71,512)(72,511)(73,510)(74,509)(75,508)(76,507)(77,506)(78,505)(79,504)(80,503)(81,502)(82,501)(83,500)(84,499)(85,498)(86,497)(87,496)(88,495)(89,494)(90,493)(91,492)(92,491)(93,490)(94,489)(95,488)(96,487)(97,486)(98,485)(99,484)(100,483)(101,482)(102,481)(103,480)(104,479)(105,478)(106,477)(107,476)(108,475)(109,474)(110,473)(111,472)(112,471)(113,470)(114,469)(115,468)(116,467)(117,466)(118,465)(119,464)(120,463)(121,462)(122,461)(123,460)(124,459)(125,458)(126,457)(127,456)(128,455)(129,454)(130,453)(131,452)(132,451)(133,450)(134,449)(135,448)(136,447)(137,446)(138,445)(139,444)(140,443)(141,442)(142,441)(143,440)(144,439)(145,438)(146,437)(147,436)(148,435)(149,434)(150,433)(151,432)(152,431)(153,430)(154,429)(155,428)(156,427)(157,426)(158,425)(159,424)(160,423)(161,422)(162,421)(163,420)(164,419)(165,418)(166,417)(167,416)(168,415)(169,414)(170,413)(171,412)(172,411)(173,410)(174,409)(175,408)(176,407)(177,406)(178,405)(179,404)(180,403)(181,402)(182,401)(183,400)(184,399)(185,398)(186,397)(187,396)(188,395)(189,394)(190,393)(191,392)(192,391)(193,390)(194,389)(195,388)(196,387)(197,386)(198,385)(199,384)(200,383)(201,382)(202,381)(203,380)(204,379)(205,378)(206,377)(207,376)(208,375)(209,374)(210,373)(211,372)(212,371)(213,370)(214,369)(215,368)(216,367)(217,366)(218,365)(219,364)(220,363)(221,362)(222,361)(223,360)(224,359)(225,358)(226,357)(227,356)(228,355)(229,354)(230,353)(231,352)(232,351)(233,350)(234,349)(235,348)(236,347)(237,346)(238,345)(239,344)(240,343)(241,342)(242,341)(243,340)(244,339)(245,338)(246,337)(247,336)(248,335)(249,334)(250,333)(251,332)(252,331)(253,330)(254,329)(255,328)(256,327)(257,326)(258,325)(259,324)(260,323)(261,322)(262,321)(263,320)(264,319)(265,318)(266,317)(267,316)(268,315)(269,314)(270,313)(271,312)(272,311)(273,310)(274,309)(275,308)(276,307)(277,306)(278,305)(279,304)(280,303)(281,302)(282,301)(283,300)(284,299)(285,298)(286,297)(287,296)(288,295)(289,294)(290,293)(291,292)

Polycyclic presentation:

2,3,5,557,2|a^2=e,b^a=b^2,c^a=c^4,d^a=d^556

Conjugacy classes & character table

8358 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33420, 8);

Dic8355 #45689205