Dic₁₃₈₁₃ #72290074

order	55252
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₁₃

Splittings

derived, Hall	C₁₉×C₇₂₇ ⋊ C₄
Sylow	C₇₂₇ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₇₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₈₁₃
commutator, hypocenter	C₁₉×C₇₂₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₉×C₇₂₇	C₂
LCS	G ▷ C₁₉×C₇₂₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 750
transitive degree	μ_t(G) = 55252

A minimal faithful permutation representation — 2 generators on 750 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750);(1,3,2,4)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(25,750)(26,749)(27,748)(28,747)(29,746)(30,745)(31,744)(32,743)(33,742)(34,741)(35,740)(36,739)(37,738)(38,737)(39,736)(40,735)(41,734)(42,733)(43,732)(44,731)(45,730)(46,729)(47,728)(48,727)(49,726)(50,725)(51,724)(52,723)(53,722)(54,721)(55,720)(56,719)(57,718)(58,717)(59,716)(60,715)(61,714)(62,713)(63,712)(64,711)(65,710)(66,709)(67,708)(68,707)(69,706)(70,705)(71,704)(72,703)(73,702)(74,701)(75,700)(76,699)(77,698)(78,697)(79,696)(80,695)(81,694)(82,693)(83,692)(84,691)(85,690)(86,689)(87,688)(88,687)(89,686)(90,685)(91,684)(92,683)(93,682)(94,681)(95,680)(96,679)(97,678)(98,677)(99,676)(100,675)(101,674)(102,673)(103,672)(104,671)(105,670)(106,669)(107,668)(108,667)(109,666)(110,665)(111,664)(112,663)(113,662)(114,661)(115,660)(116,659)(117,658)(118,657)(119,656)(120,655)(121,654)(122,653)(123,652)(124,651)(125,650)(126,649)(127,648)(128,647)(129,646)(130,645)(131,644)(132,643)(133,642)(134,641)(135,640)(136,639)(137,638)(138,637)(139,636)(140,635)(141,634)(142,633)(143,632)(144,631)(145,630)(146,629)(147,628)(148,627)(149,626)(150,625)(151,624)(152,623)(153,622)(154,621)(155,620)(156,619)(157,618)(158,617)(159,616)(160,615)(161,614)(162,613)(163,612)(164,611)(165,610)(166,609)(167,608)(168,607)(169,606)(170,605)(171,604)(172,603)(173,602)(174,601)(175,600)(176,599)(177,598)(178,597)(179,596)(180,595)(181,594)(182,593)(183,592)(184,591)(185,590)(186,589)(187,588)(188,587)(189,586)(190,585)(191,584)(192,583)(193,582)(194,581)(195,580)(196,579)(197,578)(198,577)(199,576)(200,575)(201,574)(202,573)(203,572)(204,571)(205,570)(206,569)(207,568)(208,567)(209,566)(210,565)(211,564)(212,563)(213,562)(214,561)(215,560)(216,559)(217,558)(218,557)(219,556)(220,555)(221,554)(222,553)(223,552)(224,551)(225,550)(226,549)(227,548)(228,547)(229,546)(230,545)(231,544)(232,543)(233,542)(234,541)(235,540)(236,539)(237,538)(238,537)(239,536)(240,535)(241,534)(242,533)(243,532)(244,531)(245,530)(246,529)(247,528)(248,527)(249,526)(250,525)(251,524)(252,523)(253,522)(254,521)(255,520)(256,519)(257,518)(258,517)(259,516)(260,515)(261,514)(262,513)(263,512)(264,511)(265,510)(266,509)(267,508)(268,507)(269,506)(270,505)(271,504)(272,503)(273,502)(274,501)(275,500)(276,499)(277,498)(278,497)(279,496)(280,495)(281,494)(282,493)(283,492)(284,491)(285,490)(286,489)(287,488)(288,487)(289,486)(290,485)(291,484)(292,483)(293,482)(294,481)(295,480)(296,479)(297,478)(298,477)(299,476)(300,475)(301,474)(302,473)(303,472)(304,471)(305,470)(306,469)(307,468)(308,467)(309,466)(310,465)(311,464)(312,463)(313,462)(314,461)(315,460)(316,459)(317,458)(318,457)(319,456)(320,455)(321,454)(322,453)(323,452)(324,451)(325,450)(326,449)(327,448)(328,447)(329,446)(330,445)(331,444)(332,443)(333,442)(334,441)(335,440)(336,439)(337,438)(338,437)(339,436)(340,435)(341,434)(342,433)(343,432)(344,431)(345,430)(346,429)(347,428)(348,427)(349,426)(350,425)(351,424)(352,423)(353,422)(354,421)(355,420)(356,419)(357,418)(358,417)(359,416)(360,415)(361,414)(362,413)(363,412)(364,411)(365,410)(366,409)(367,408)(368,407)(369,406)(370,405)(371,404)(372,403)(373,402)(374,401)(375,400)(376,399)(377,398)(378,397)(379,396)(380,395)(381,394)(382,393)(383,392)(384,391)(385,390)(386,389)(387,388)

Polycyclic presentation:

2^2,19,727|a^2=b,c^a=c^18,d^a=d^726

Conjugacy classes & character table

13816 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d");;
gap> coll := SingleCollector(F, [2, 2, 19, 727]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^18);;
gap> SetConjugate(coll, 4, 1, F.4^726);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13813 #72290074