Dic₅₇₅₃ #37527673

order	23012
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₇₅₃
unambiguous semidirect product	C₁₁⋊Dic₅₂₃
unambiguous semidirect product	C₅₂₃⋊Dic₁₁

Splittings

derived, Hall	C₁₁×C₅₂₃ ⋊ C₄
Sylow	C₅₂₃ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₅₂₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₇₅₃
commutator, hypocenter	C₁₁×C₅₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₅₂₃	C₂
LCS	G ▷ C₁₁×C₅₂₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 538
transitive degree	μ_t(G) = 23012

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

Polycyclic presentation:

2,11,523,2|a^2=d,b^a=b^10,c^a=c^522

Conjugacy classes & character table

5756 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₅₀₆
maximal normal subgroup of	Dic₁₁₅₀₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23012, 4);

Dic5753 #37527673