Dic₅₅₇₀ #72224115

order	22280
automorphism	Aut(G) order 49550720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₅₅₇₀

Splittings

Hall	C₅×C₅₅₇ ⋊ Q₈
Sylow	C₅₅₇ ⋊ Dic₁₀
Sylow	C₅ ⋊ Dic₁₁₁₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₇₈₅
commutator, socle, prime radical, cosocle	C₂×C₅×C₅₅₇	C₂²
Fitting	C₄×C₅×C₅₅₇	C₂
hypercenter	C₄	D₂₇₈₅
hypocenter	C₅×C₅₅₇	Q₈
LCS	G ▷ C₂×C₅×C₅₅₇ ▷ C₅×C₅₅₇
UCS	C₁ ◁ C₂ ◁ C₄

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 570
transitive degree	μ_t(G) = 22280

A minimal faithful permutation representation — 2 generators on 570 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);(1,5,3,7)(2,6,4,8)(10,13)(11,12)(15,570)(16,569)(17,568)(18,567)(19,566)(20,565)(21,564)(22,563)(23,562)(24,561)(25,560)(26,559)(27,558)(28,557)(29,556)(30,555)(31,554)(32,553)(33,552)(34,551)(35,550)(36,549)(37,548)(38,547)(39,546)(40,545)(41,544)(42,543)(43,542)(44,541)(45,540)(46,539)(47,538)(48,537)(49,536)(50,535)(51,534)(52,533)(53,532)(54,531)(55,530)(56,529)(57,528)(58,527)(59,526)(60,525)(61,524)(62,523)(63,522)(64,521)(65,520)(66,519)(67,518)(68,517)(69,516)(70,515)(71,514)(72,513)(73,512)(74,511)(75,510)(76,509)(77,508)(78,507)(79,506)(80,505)(81,504)(82,503)(83,502)(84,501)(85,500)(86,499)(87,498)(88,497)(89,496)(90,495)(91,494)(92,493)(93,492)(94,491)(95,490)(96,489)(97,488)(98,487)(99,486)(100,485)(101,484)(102,483)(103,482)(104,481)(105,480)(106,479)(107,478)(108,477)(109,476)(110,475)(111,474)(112,473)(113,472)(114,471)(115,470)(116,469)(117,468)(118,467)(119,466)(120,465)(121,464)(122,463)(123,462)(124,461)(125,460)(126,459)(127,458)(128,457)(129,456)(130,455)(131,454)(132,453)(133,452)(134,451)(135,450)(136,449)(137,448)(138,447)(139,446)(140,445)(141,444)(142,443)(143,442)(144,441)(145,440)(146,439)(147,438)(148,437)(149,436)(150,435)(151,434)(152,433)(153,432)(154,431)(155,430)(156,429)(157,428)(158,427)(159,426)(160,425)(161,424)(162,423)(163,422)(164,421)(165,420)(166,419)(167,418)(168,417)(169,416)(170,415)(171,414)(172,413)(173,412)(174,411)(175,410)(176,409)(177,408)(178,407)(179,406)(180,405)(181,404)(182,403)(183,402)(184,401)(185,400)(186,399)(187,398)(188,397)(189,396)(190,395)(191,394)(192,393)(193,392)(194,391)(195,390)(196,389)(197,388)(198,387)(199,386)(200,385)(201,384)(202,383)(203,382)(204,381)(205,380)(206,379)(207,378)(208,377)(209,376)(210,375)(211,374)(212,373)(213,372)(214,371)(215,370)(216,369)(217,368)(218,367)(219,366)(220,365)(221,364)(222,363)(223,362)(224,361)(225,360)(226,359)(227,358)(228,357)(229,356)(230,355)(231,354)(232,353)(233,352)(234,351)(235,350)(236,349)(237,348)(238,347)(239,346)(240,345)(241,344)(242,343)(243,342)(244,341)(245,340)(246,339)(247,338)(248,337)(249,336)(250,335)(251,334)(252,333)(253,332)(254,331)(255,330)(256,329)(257,328)(258,327)(259,326)(260,325)(261,324)(262,323)(263,322)(264,321)(265,320)(266,319)(267,318)(268,317)(269,316)(270,315)(271,314)(272,313)(273,312)(274,311)(275,310)(276,309)(277,308)(278,307)(279,306)(280,305)(281,304)(282,303)(283,302)(284,301)(285,300)(286,299)(287,298)(288,297)(289,296)(290,295)(291,294)(292,293)

Polycyclic presentation:

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

Conjugacy classes & character table

5573 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₁₄₀
maximal normal subgroup of	Dic₁₁₁₄₀

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, 5, 557]);;
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.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^556);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic5570 #72224115