Dic₆₃₈₁ #38784198

order	25524
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₃₈₁
unambiguous semidirect product	C₉⋊Dic₇₀₉

Splittings

derived, Hall	C₉×C₇₀₉ ⋊ C₄
Sylow	C₇₀₉ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₇₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₃₈₁
commutator, hypocenter	C₉×C₇₀₉	C₄
Frattini	C₂×C₃	D₂₁₂₇
Fitting, cosocle	C₂×C₉×C₇₀₉	C₂
socle, prime radical	C₂×C₃×C₇₀₉	D₃
LCS	G ▷ C₉×C₇₀₉
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 722
transitive degree	μ_t(G) = 25524

A minimal faithful permutation representation — 2 generators on 722 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,722)(16,721)(17,720)(18,719)(19,718)(20,717)(21,716)(22,715)(23,714)(24,713)(25,712)(26,711)(27,710)(28,709)(29,708)(30,707)(31,706)(32,705)(33,704)(34,703)(35,702)(36,701)(37,700)(38,699)(39,698)(40,697)(41,696)(42,695)(43,694)(44,693)(45,692)(46,691)(47,690)(48,689)(49,688)(50,687)(51,686)(52,685)(53,684)(54,683)(55,682)(56,681)(57,680)(58,679)(59,678)(60,677)(61,676)(62,675)(63,674)(64,673)(65,672)(66,671)(67,670)(68,669)(69,668)(70,667)(71,666)(72,665)(73,664)(74,663)(75,662)(76,661)(77,660)(78,659)(79,658)(80,657)(81,656)(82,655)(83,654)(84,653)(85,652)(86,651)(87,650)(88,649)(89,648)(90,647)(91,646)(92,645)(93,644)(94,643)(95,642)(96,641)(97,640)(98,639)(99,638)(100,637)(101,636)(102,635)(103,634)(104,633)(105,632)(106,631)(107,630)(108,629)(109,628)(110,627)(111,626)(112,625)(113,624)(114,623)(115,622)(116,621)(117,620)(118,619)(119,618)(120,617)(121,616)(122,615)(123,614)(124,613)(125,612)(126,611)(127,610)(128,609)(129,608)(130,607)(131,606)(132,605)(133,604)(134,603)(135,602)(136,601)(137,600)(138,599)(139,598)(140,597)(141,596)(142,595)(143,594)(144,593)(145,592)(146,591)(147,590)(148,589)(149,588)(150,587)(151,586)(152,585)(153,584)(154,583)(155,582)(156,581)(157,580)(158,579)(159,578)(160,577)(161,576)(162,575)(163,574)(164,573)(165,572)(166,571)(167,570)(168,569)(169,568)(170,567)(171,566)(172,565)(173,564)(174,563)(175,562)(176,561)(177,560)(178,559)(179,558)(180,557)(181,556)(182,555)(183,554)(184,553)(185,552)(186,551)(187,550)(188,549)(189,548)(190,547)(191,546)(192,545)(193,544)(194,543)(195,542)(196,541)(197,540)(198,539)(199,538)(200,537)(201,536)(202,535)(203,534)(204,533)(205,532)(206,531)(207,530)(208,529)(209,528)(210,527)(211,526)(212,525)(213,524)(214,523)(215,522)(216,521)(217,520)(218,519)(219,518)(220,517)(221,516)(222,515)(223,514)(224,513)(225,512)(226,511)(227,510)(228,509)(229,508)(230,507)(231,506)(232,505)(233,504)(234,503)(235,502)(236,501)(237,500)(238,499)(239,498)(240,497)(241,496)(242,495)(243,494)(244,493)(245,492)(246,491)(247,490)(248,489)(249,488)(250,487)(251,486)(252,485)(253,484)(254,483)(255,482)(256,481)(257,480)(258,479)(259,478)(260,477)(261,476)(262,475)(263,474)(264,473)(265,472)(266,471)(267,470)(268,469)(269,468)(270,467)(271,466)(272,465)(273,464)(274,463)(275,462)(276,461)(277,460)(278,459)(279,458)(280,457)(281,456)(282,455)(283,454)(284,453)(285,452)(286,451)(287,450)(288,449)(289,448)(290,447)(291,446)(292,445)(293,444)(294,443)(295,442)(296,441)(297,440)(298,439)(299,438)(300,437)(301,436)(302,435)(303,434)(304,433)(305,432)(306,431)(307,430)(308,429)(309,428)(310,427)(311,426)(312,425)(313,424)(314,423)(315,422)(316,421)(317,420)(318,419)(319,418)(320,417)(321,416)(322,415)(323,414)(324,413)(325,412)(326,411)(327,410)(328,409)(329,408)(330,407)(331,406)(332,405)(333,404)(334,403)(335,402)(336,401)(337,400)(338,399)(339,398)(340,397)(341,396)(342,395)(343,394)(344,393)(345,392)(346,391)(347,390)(348,389)(349,388)(350,387)(351,386)(352,385)(353,384)(354,383)(355,382)(356,381)(357,380)(358,379)(359,378)(360,377)(361,376)(362,375)(363,374)(364,373)(365,372)(366,371)(367,370)(368,369)

Polycyclic presentation:

2,3,709,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^708,e^a=e^2

Conjugacy classes & character table

6384 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₇₆₂
maximal normal subgroup of	Dic₁₂₇₆₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25524, 6);

Dic6381 #38784198