Dic₆₁₄₇ #37589645

order	24588
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₁₄₇
unambiguous semidirect product	C₉⋊Dic₆₈₃
unambiguous semidirect product	C₆₈₃⋊Dic₉

Splittings

derived, Hall	C₉×C₆₈₃ ⋊ C₄
Sylow	C₆₈₃ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₆₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₁₄₇
commutator, hypocenter	C₉×C₆₈₃	C₄
Frattini	C₂×C₃	D₂₀₄₉
Fitting, cosocle	C₂×C₉×C₆₈₃	C₂
socle, prime radical	C₂×C₃×C₆₈₃	D₃
LCS	G ▷ C₉×C₆₈₃
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 696
transitive degree	μ_t(G) = 24588

A minimal faithful permutation representation — 2 generators on 696 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,696)(16,695)(17,694)(18,693)(19,692)(20,691)(21,690)(22,689)(23,688)(24,687)(25,686)(26,685)(27,684)(28,683)(29,682)(30,681)(31,680)(32,679)(33,678)(34,677)(35,676)(36,675)(37,674)(38,673)(39,672)(40,671)(41,670)(42,669)(43,668)(44,667)(45,666)(46,665)(47,664)(48,663)(49,662)(50,661)(51,660)(52,659)(53,658)(54,657)(55,656)(56,655)(57,654)(58,653)(59,652)(60,651)(61,650)(62,649)(63,648)(64,647)(65,646)(66,645)(67,644)(68,643)(69,642)(70,641)(71,640)(72,639)(73,638)(74,637)(75,636)(76,635)(77,634)(78,633)(79,632)(80,631)(81,630)(82,629)(83,628)(84,627)(85,626)(86,625)(87,624)(88,623)(89,622)(90,621)(91,620)(92,619)(93,618)(94,617)(95,616)(96,615)(97,614)(98,613)(99,612)(100,611)(101,610)(102,609)(103,608)(104,607)(105,606)(106,605)(107,604)(108,603)(109,602)(110,601)(111,600)(112,599)(113,598)(114,597)(115,596)(116,595)(117,594)(118,593)(119,592)(120,591)(121,590)(122,589)(123,588)(124,587)(125,586)(126,585)(127,584)(128,583)(129,582)(130,581)(131,580)(132,579)(133,578)(134,577)(135,576)(136,575)(137,574)(138,573)(139,572)(140,571)(141,570)(142,569)(143,568)(144,567)(145,566)(146,565)(147,564)(148,563)(149,562)(150,561)(151,560)(152,559)(153,558)(154,557)(155,556)(156,555)(157,554)(158,553)(159,552)(160,551)(161,550)(162,549)(163,548)(164,547)(165,546)(166,545)(167,544)(168,543)(169,542)(170,541)(171,540)(172,539)(173,538)(174,537)(175,536)(176,535)(177,534)(178,533)(179,532)(180,531)(181,530)(182,529)(183,528)(184,527)(185,526)(186,525)(187,524)(188,523)(189,522)(190,521)(191,520)(192,519)(193,518)(194,517)(195,516)(196,515)(197,514)(198,513)(199,512)(200,511)(201,510)(202,509)(203,508)(204,507)(205,506)(206,505)(207,504)(208,503)(209,502)(210,501)(211,500)(212,499)(213,498)(214,497)(215,496)(216,495)(217,494)(218,493)(219,492)(220,491)(221,490)(222,489)(223,488)(224,487)(225,486)(226,485)(227,484)(228,483)(229,482)(230,481)(231,480)(232,479)(233,478)(234,477)(235,476)(236,475)(237,474)(238,473)(239,472)(240,471)(241,470)(242,469)(243,468)(244,467)(245,466)(246,465)(247,464)(248,463)(249,462)(250,461)(251,460)(252,459)(253,458)(254,457)(255,456)(256,455)(257,454)(258,453)(259,452)(260,451)(261,450)(262,449)(263,448)(264,447)(265,446)(266,445)(267,444)(268,443)(269,442)(270,441)(271,440)(272,439)(273,438)(274,437)(275,436)(276,435)(277,434)(278,433)(279,432)(280,431)(281,430)(282,429)(283,428)(284,427)(285,426)(286,425)(287,424)(288,423)(289,422)(290,421)(291,420)(292,419)(293,418)(294,417)(295,416)(296,415)(297,414)(298,413)(299,412)(300,411)(301,410)(302,409)(303,408)(304,407)(305,406)(306,405)(307,404)(308,403)(309,402)(310,401)(311,400)(312,399)(313,398)(314,397)(315,396)(316,395)(317,394)(318,393)(319,392)(320,391)(321,390)(322,389)(323,388)(324,387)(325,386)(326,385)(327,384)(328,383)(329,382)(330,381)(331,380)(332,379)(333,378)(334,377)(335,376)(336,375)(337,374)(338,373)(339,372)(340,371)(341,370)(342,369)(343,368)(344,367)(345,366)(346,365)(347,364)(348,363)(349,362)(350,361)(351,360)(352,359)(353,358)(354,357)(355,356)

Polycyclic presentation:

2,3,683,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^682,e^a=e^2

Conjugacy classes & character table

6150 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₂₉₄
maximal normal subgroup of	Dic₁₂₂₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24588, 4);

Dic6147 #37589645