Dic₆₅₄₃ #39908646

order	26172
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₅₄₃
unambiguous semidirect product	C₉⋊Dic₇₂₇
unambiguous semidirect product	C₇₂₇⋊Dic₉

Splittings

derived, Hall	C₉×C₇₂₇ ⋊ C₄
Sylow	C₇₂₇ ⋊ Dic₉
Sylow	C₉ ⋊ Dic₇₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₆₅₄₃
commutator, hypocenter	C₉×C₇₂₇	C₄
Frattini	C₂×C₃	D₂₁₈₁
Fitting, cosocle	C₂×C₉×C₇₂₇	C₂
socle, prime radical	C₂×C₃×C₇₂₇	D₃
LCS	G ▷ C₉×C₇₂₇
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 740
transitive degree	μ_t(G) = 26172

A minimal faithful permutation representation — 2 generators on 740 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,740)(16,739)(17,738)(18,737)(19,736)(20,735)(21,734)(22,733)(23,732)(24,731)(25,730)(26,729)(27,728)(28,727)(29,726)(30,725)(31,724)(32,723)(33,722)(34,721)(35,720)(36,719)(37,718)(38,717)(39,716)(40,715)(41,714)(42,713)(43,712)(44,711)(45,710)(46,709)(47,708)(48,707)(49,706)(50,705)(51,704)(52,703)(53,702)(54,701)(55,700)(56,699)(57,698)(58,697)(59,696)(60,695)(61,694)(62,693)(63,692)(64,691)(65,690)(66,689)(67,688)(68,687)(69,686)(70,685)(71,684)(72,683)(73,682)(74,681)(75,680)(76,679)(77,678)(78,677)(79,676)(80,675)(81,674)(82,673)(83,672)(84,671)(85,670)(86,669)(87,668)(88,667)(89,666)(90,665)(91,664)(92,663)(93,662)(94,661)(95,660)(96,659)(97,658)(98,657)(99,656)(100,655)(101,654)(102,653)(103,652)(104,651)(105,650)(106,649)(107,648)(108,647)(109,646)(110,645)(111,644)(112,643)(113,642)(114,641)(115,640)(116,639)(117,638)(118,637)(119,636)(120,635)(121,634)(122,633)(123,632)(124,631)(125,630)(126,629)(127,628)(128,627)(129,626)(130,625)(131,624)(132,623)(133,622)(134,621)(135,620)(136,619)(137,618)(138,617)(139,616)(140,615)(141,614)(142,613)(143,612)(144,611)(145,610)(146,609)(147,608)(148,607)(149,606)(150,605)(151,604)(152,603)(153,602)(154,601)(155,600)(156,599)(157,598)(158,597)(159,596)(160,595)(161,594)(162,593)(163,592)(164,591)(165,590)(166,589)(167,588)(168,587)(169,586)(170,585)(171,584)(172,583)(173,582)(174,581)(175,580)(176,579)(177,578)(178,577)(179,576)(180,575)(181,574)(182,573)(183,572)(184,571)(185,570)(186,569)(187,568)(188,567)(189,566)(190,565)(191,564)(192,563)(193,562)(194,561)(195,560)(196,559)(197,558)(198,557)(199,556)(200,555)(201,554)(202,553)(203,552)(204,551)(205,550)(206,549)(207,548)(208,547)(209,546)(210,545)(211,544)(212,543)(213,542)(214,541)(215,540)(216,539)(217,538)(218,537)(219,536)(220,535)(221,534)(222,533)(223,532)(224,531)(225,530)(226,529)(227,528)(228,527)(229,526)(230,525)(231,524)(232,523)(233,522)(234,521)(235,520)(236,519)(237,518)(238,517)(239,516)(240,515)(241,514)(242,513)(243,512)(244,511)(245,510)(246,509)(247,508)(248,507)(249,506)(250,505)(251,504)(252,503)(253,502)(254,501)(255,500)(256,499)(257,498)(258,497)(259,496)(260,495)(261,494)(262,493)(263,492)(264,491)(265,490)(266,489)(267,488)(268,487)(269,486)(270,485)(271,484)(272,483)(273,482)(274,481)(275,480)(276,479)(277,478)(278,477)(279,476)(280,475)(281,474)(282,473)(283,472)(284,471)(285,470)(286,469)(287,468)(288,467)(289,466)(290,465)(291,464)(292,463)(293,462)(294,461)(295,460)(296,459)(297,458)(298,457)(299,456)(300,455)(301,454)(302,453)(303,452)(304,451)(305,450)(306,449)(307,448)(308,447)(309,446)(310,445)(311,444)(312,443)(313,442)(314,441)(315,440)(316,439)(317,438)(318,437)(319,436)(320,435)(321,434)(322,433)(323,432)(324,431)(325,430)(326,429)(327,428)(328,427)(329,426)(330,425)(331,424)(332,423)(333,422)(334,421)(335,420)(336,419)(337,418)(338,417)(339,416)(340,415)(341,414)(342,413)(343,412)(344,411)(345,410)(346,409)(347,408)(348,407)(349,406)(350,405)(351,404)(352,403)(353,402)(354,401)(355,400)(356,399)(357,398)(358,397)(359,396)(360,395)(361,394)(362,393)(363,392)(364,391)(365,390)(366,389)(367,388)(368,387)(369,386)(370,385)(371,384)(372,383)(373,382)(374,381)(375,380)(376,379)(377,378)

Polycyclic presentation:

2,3,727,2,3|a^2=d,b^3=e^2,b^a=b^2e,c^a=c^726,e^a=e^2

Conjugacy classes & character table

6546 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₀₈₆
maximal normal subgroup of	Dic₁₃₀₈₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26172, 6);

Dic6543 #39908646