D₁₃₁₂₉ #39910943

order	26258
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₁₂₉
unambiguous semidirect product	C₁₉⋊D₆₉₁
unambiguous semidirect product	C₆₉₁⋊D₁₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₆₉₁ ⋊ C₂
Sylow	C₆₉₁ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₆₉₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₆₉₁	C₂
LCS	G ▷ C₁₉×C₆₉₁
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 710
transitive degree	μ_t(G) = 13129

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

Polycyclic presentation:

2,19,691|b^a=b^18,c^a=c^690

Conjugacy classes & character table

6566 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₁₂₉
maximal normal subgroup of	C₂×D₁₃₁₂₉
maximal quotient of	Dic₁₃₁₂₉, C₂×D₁₃₁₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26258, 3);

D13129 #39910943