Dic₆₀₁₇ #37562701

order	24068
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₁₇
unambiguous semidirect product	C₁₁⋊Dic₅₄₇
unambiguous semidirect product	C₅₄₇⋊Dic₁₁

Splittings

derived, Hall	C₁₁×C₅₄₇ ⋊ C₄
Sylow	C₅₄₇ ⋊ Dic₁₁
Sylow	C₁₁ ⋊ Dic₅₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₀₁₇
commutator, hypocenter	C₁₁×C₅₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₅₄₇	C₂
LCS	G ▷ C₁₁×C₅₄₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 562
transitive degree	μ_t(G) = 24068

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

Polycyclic presentation:

2,11,547,2|a^2=d,b^a=b^10,c^a=c^546

Conjugacy classes & character table

6020 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₀₃₄
maximal normal subgroup of	Dic₁₂₀₃₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24068, 4);

Dic6017 #37562701