Dic₅₉₅₁ #37558352

order	23804
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₉₅₁
unambiguous semidirect product	C₁₁⋊Dic₅₄₁

Splittings

derived, Hall	C₁₁×C₅₄₁ ⋊ C₄
Sylow	C₅₄₁ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₅₄₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₉₅₁
commutator, hypocenter	C₁₁×C₅₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₅₄₁	C₂
LCS	G ▷ C₁₁×C₅₄₁
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 556
transitive degree	μ_t(G) = 23804

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

Polycyclic presentation:

2,11,541,2|a^2=d,b^a=b^10,c^a=c^540

Conjugacy classes & character table

5954 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₉₀₂
maximal normal subgroup of	Dic₁₁₉₀₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23804, 4);

Dic5951 #37558352