Dic₁₃₈₅₁ #72290146

order	55404
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₅₁

Splittings

derived, Hall	C₁₉×C₇₂₉ ⋊ C₄
Sylow	C₇₂₉ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₇₂₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₈₅₁
commutator, hypocenter	C₁₉×C₇₂₉	C₄
Frattini	C₂×C₂₄₃	D₅₇
Fitting, cosocle	C₂×C₁₉×C₇₂₉	C₂
socle, prime radical	C₂×C₃×C₁₉	D₂₄₃
LCS	G ▷ C₁₉×C₇₂₉
UCS	C₁ ◁ C₂

42 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 752
transitive degree	μ_t(G) = 55404

A minimal faithful permutation representation — 2 generators on 752 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)(734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752);(1,3,2,4)(6,733)(7,732)(8,731)(9,730)(10,729)(11,728)(12,727)(13,726)(14,725)(15,724)(16,723)(17,722)(18,721)(19,720)(20,719)(21,718)(22,717)(23,716)(24,715)(25,714)(26,713)(27,712)(28,711)(29,710)(30,709)(31,708)(32,707)(33,706)(34,705)(35,704)(36,703)(37,702)(38,701)(39,700)(40,699)(41,698)(42,697)(43,696)(44,695)(45,694)(46,693)(47,692)(48,691)(49,690)(50,689)(51,688)(52,687)(53,686)(54,685)(55,684)(56,683)(57,682)(58,681)(59,680)(60,679)(61,678)(62,677)(63,676)(64,675)(65,674)(66,673)(67,672)(68,671)(69,670)(70,669)(71,668)(72,667)(73,666)(74,665)(75,664)(76,663)(77,662)(78,661)(79,660)(80,659)(81,658)(82,657)(83,656)(84,655)(85,654)(86,653)(87,652)(88,651)(89,650)(90,649)(91,648)(92,647)(93,646)(94,645)(95,644)(96,643)(97,642)(98,641)(99,640)(100,639)(101,638)(102,637)(103,636)(104,635)(105,634)(106,633)(107,632)(108,631)(109,630)(110,629)(111,628)(112,627)(113,626)(114,625)(115,624)(116,623)(117,622)(118,621)(119,620)(120,619)(121,618)(122,617)(123,616)(124,615)(125,614)(126,613)(127,612)(128,611)(129,610)(130,609)(131,608)(132,607)(133,606)(134,605)(135,604)(136,603)(137,602)(138,601)(139,600)(140,599)(141,598)(142,597)(143,596)(144,595)(145,594)(146,593)(147,592)(148,591)(149,590)(150,589)(151,588)(152,587)(153,586)(154,585)(155,584)(156,583)(157,582)(158,581)(159,580)(160,579)(161,578)(162,577)(163,576)(164,575)(165,574)(166,573)(167,572)(168,571)(169,570)(170,569)(171,568)(172,567)(173,566)(174,565)(175,564)(176,563)(177,562)(178,561)(179,560)(180,559)(181,558)(182,557)(183,556)(184,555)(185,554)(186,553)(187,552)(188,551)(189,550)(190,549)(191,548)(192,547)(193,546)(194,545)(195,544)(196,543)(197,542)(198,541)(199,540)(200,539)(201,538)(202,537)(203,536)(204,535)(205,534)(206,533)(207,532)(208,531)(209,530)(210,529)(211,528)(212,527)(213,526)(214,525)(215,524)(216,523)(217,522)(218,521)(219,520)(220,519)(221,518)(222,517)(223,516)(224,515)(225,514)(226,513)(227,512)(228,511)(229,510)(230,509)(231,508)(232,507)(233,506)(234,505)(235,504)(236,503)(237,502)(238,501)(239,500)(240,499)(241,498)(242,497)(243,496)(244,495)(245,494)(246,493)(247,492)(248,491)(249,490)(250,489)(251,488)(252,487)(253,486)(254,485)(255,484)(256,483)(257,482)(258,481)(259,480)(260,479)(261,478)(262,477)(263,476)(264,475)(265,474)(266,473)(267,472)(268,471)(269,470)(270,469)(271,468)(272,467)(273,466)(274,465)(275,464)(276,463)(277,462)(278,461)(279,460)(280,459)(281,458)(282,457)(283,456)(284,455)(285,454)(286,453)(287,452)(288,451)(289,450)(290,449)(291,448)(292,447)(293,446)(294,445)(295,444)(296,443)(297,442)(298,441)(299,440)(300,439)(301,438)(302,437)(303,436)(304,435)(305,434)(306,433)(307,432)(308,431)(309,430)(310,429)(311,428)(312,427)(313,426)(314,425)(315,424)(316,423)(317,422)(318,421)(319,420)(320,419)(321,418)(322,417)(323,416)(324,415)(325,414)(326,413)(327,412)(328,411)(329,410)(330,409)(331,408)(332,407)(333,406)(334,405)(335,404)(336,403)(337,402)(338,401)(339,400)(340,399)(341,398)(342,397)(343,396)(344,395)(345,394)(346,393)(347,392)(348,391)(349,390)(350,389)(351,388)(352,387)(353,386)(354,385)(355,384)(356,383)(357,382)(358,381)(359,380)(360,379)(361,378)(362,377)(363,376)(364,375)(365,374)(366,373)(367,372)(368,371)(369,370)(735,752)(736,751)(737,750)(738,749)(739,748)(740,747)(741,746)(742,745)(743,744)

Polycyclic presentation:

2^2,3,19,3^5|a^2=b,c^3=e,e^3=f,f^3=g,g^3=h,h^3=i,c^a=c^2e^2f^2g^2h^2i^2,d^a=d^18,e^a=e^2f^2g^2h^2i^2,f^a=f^2g^2h^2i^2,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

13854 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", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 3, 19, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 3, 1, F.3^2*F.5^2*F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 4, 1, F.4^18);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13851 #72290146