Dic₆₀₉₃ #37567111

order	24372
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₉₃
unambiguous semidirect product	C₉⋊Dic₆₇₇

Splittings

derived, Hall	C₉×C₆₇₇ ⋊ C₄
Sylow	C₆₇₇ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₆₇₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₀₉₃
commutator, hypocenter	C₉×C₆₇₇	C₄
Frattini	C₂×C₃	D₂₀₃₁
Fitting, cosocle	C₂×C₉×C₆₇₇	C₂
socle, prime radical	C₂×C₃×C₆₇₇	D₃
LCS	G ▷ C₉×C₆₇₇
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 690
transitive degree	μ_t(G) = 24372

A minimal faithful permutation representation — 2 generators on 690 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,690)(16,689)(17,688)(18,687)(19,686)(20,685)(21,684)(22,683)(23,682)(24,681)(25,680)(26,679)(27,678)(28,677)(29,676)(30,675)(31,674)(32,673)(33,672)(34,671)(35,670)(36,669)(37,668)(38,667)(39,666)(40,665)(41,664)(42,663)(43,662)(44,661)(45,660)(46,659)(47,658)(48,657)(49,656)(50,655)(51,654)(52,653)(53,652)(54,651)(55,650)(56,649)(57,648)(58,647)(59,646)(60,645)(61,644)(62,643)(63,642)(64,641)(65,640)(66,639)(67,638)(68,637)(69,636)(70,635)(71,634)(72,633)(73,632)(74,631)(75,630)(76,629)(77,628)(78,627)(79,626)(80,625)(81,624)(82,623)(83,622)(84,621)(85,620)(86,619)(87,618)(88,617)(89,616)(90,615)(91,614)(92,613)(93,612)(94,611)(95,610)(96,609)(97,608)(98,607)(99,606)(100,605)(101,604)(102,603)(103,602)(104,601)(105,600)(106,599)(107,598)(108,597)(109,596)(110,595)(111,594)(112,593)(113,592)(114,591)(115,590)(116,589)(117,588)(118,587)(119,586)(120,585)(121,584)(122,583)(123,582)(124,581)(125,580)(126,579)(127,578)(128,577)(129,576)(130,575)(131,574)(132,573)(133,572)(134,571)(135,570)(136,569)(137,568)(138,567)(139,566)(140,565)(141,564)(142,563)(143,562)(144,561)(145,560)(146,559)(147,558)(148,557)(149,556)(150,555)(151,554)(152,553)(153,552)(154,551)(155,550)(156,549)(157,548)(158,547)(159,546)(160,545)(161,544)(162,543)(163,542)(164,541)(165,540)(166,539)(167,538)(168,537)(169,536)(170,535)(171,534)(172,533)(173,532)(174,531)(175,530)(176,529)(177,528)(178,527)(179,526)(180,525)(181,524)(182,523)(183,522)(184,521)(185,520)(186,519)(187,518)(188,517)(189,516)(190,515)(191,514)(192,513)(193,512)(194,511)(195,510)(196,509)(197,508)(198,507)(199,506)(200,505)(201,504)(202,503)(203,502)(204,501)(205,500)(206,499)(207,498)(208,497)(209,496)(210,495)(211,494)(212,493)(213,492)(214,491)(215,490)(216,489)(217,488)(218,487)(219,486)(220,485)(221,484)(222,483)(223,482)(224,481)(225,480)(226,479)(227,478)(228,477)(229,476)(230,475)(231,474)(232,473)(233,472)(234,471)(235,470)(236,469)(237,468)(238,467)(239,466)(240,465)(241,464)(242,463)(243,462)(244,461)(245,460)(246,459)(247,458)(248,457)(249,456)(250,455)(251,454)(252,453)(253,452)(254,451)(255,450)(256,449)(257,448)(258,447)(259,446)(260,445)(261,444)(262,443)(263,442)(264,441)(265,440)(266,439)(267,438)(268,437)(269,436)(270,435)(271,434)(272,433)(273,432)(274,431)(275,430)(276,429)(277,428)(278,427)(279,426)(280,425)(281,424)(282,423)(283,422)(284,421)(285,420)(286,419)(287,418)(288,417)(289,416)(290,415)(291,414)(292,413)(293,412)(294,411)(295,410)(296,409)(297,408)(298,407)(299,406)(300,405)(301,404)(302,403)(303,402)(304,401)(305,400)(306,399)(307,398)(308,397)(309,396)(310,395)(311,394)(312,393)(313,392)(314,391)(315,390)(316,389)(317,388)(318,387)(319,386)(320,385)(321,384)(322,383)(323,382)(324,381)(325,380)(326,379)(327,378)(328,377)(329,376)(330,375)(331,374)(332,373)(333,372)(334,371)(335,370)(336,369)(337,368)(338,367)(339,366)(340,365)(341,364)(342,363)(343,362)(344,361)(345,360)(346,359)(347,358)(348,357)(349,356)(350,355)(351,354)(352,353)

Polycyclic presentation:

2,3,677,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^676,e^a=e^2

Conjugacy classes & character table

6096 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₁₈₆
maximal normal subgroup of	Dic₁₂₁₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24372, 4);

Dic6093 #37567111