Dic₅₇₃₁ #37523980

order	22924
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇₃₁
unambiguous semidirect product	C₁₁⋊Dic₅₂₁

Splittings

derived, Hall	C₁₁×C₅₂₁ ⋊ C₄
Sylow	C₅₂₁ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₅₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₃₁
commutator, hypocenter	C₁₁×C₅₂₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₅₂₁	C₂
LCS	G ▷ C₁₁×C₅₂₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 536
transitive degree	μ_t(G) = 22924

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

Polycyclic presentation:

2,11,521,2|a^2=d,b^a=b^10,c^a=c^520

Conjugacy classes & character table

5734 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₄₆₂
maximal normal subgroup of	Dic₁₁₄₆₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22924, 4);

Dic5731 #37523980