D₁₃₈₁₃ #42120521

order	27626
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₈₁₃
unambiguous semidirect product	C₁₉⋊D₇₂₇
unambiguous semidirect product	C₇₂₇⋊D₁₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₇₂₇ ⋊ C₂
Sylow	C₇₂₇ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₇₂₇

Subgroups

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

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 746
transitive degree	μ_t(G) = 13813

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

Polycyclic presentation:

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

Conjugacy classes & character table

6908 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₈₁₃
maximal normal subgroup of	C₂×D₁₃₈₁₃
maximal quotient of	Dic₁₃₈₁₃, C₂×D₁₃₈₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27626, 3);

D13813 #42120521