D₁₅₇₇₁ #44538724

order	31542
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₁₅₇₇₁

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₇₅₁ ⋊ C₂
Hall	C₇×C₇₅₁ ⋊ D₃
Hall	C₃×C₇₅₁ ⋊ D₇
Sylow	C₇₅₁ ⋊ D₂₁
Hall	C₃×C₇ ⋊ D₇₅₁
Sylow	C₇ ⋊ D₂₂₅₃
Sylow	C₃ ⋊ D₅₂₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₇₅₁	C₂
LCS	G ▷ C₃×C₇×C₇₅₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 761
transitive degree	μ_t(G) = 15771

A minimal faithful permutation representation — 2 generators on 761 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761);(2,3)(5,10)(6,9)(7,8)(12,761)(13,760)(14,759)(15,758)(16,757)(17,756)(18,755)(19,754)(20,753)(21,752)(22,751)(23,750)(24,749)(25,748)(26,747)(27,746)(28,745)(29,744)(30,743)(31,742)(32,741)(33,740)(34,739)(35,738)(36,737)(37,736)(38,735)(39,734)(40,733)(41,732)(42,731)(43,730)(44,729)(45,728)(46,727)(47,726)(48,725)(49,724)(50,723)(51,722)(52,721)(53,720)(54,719)(55,718)(56,717)(57,716)(58,715)(59,714)(60,713)(61,712)(62,711)(63,710)(64,709)(65,708)(66,707)(67,706)(68,705)(69,704)(70,703)(71,702)(72,701)(73,700)(74,699)(75,698)(76,697)(77,696)(78,695)(79,694)(80,693)(81,692)(82,691)(83,690)(84,689)(85,688)(86,687)(87,686)(88,685)(89,684)(90,683)(91,682)(92,681)(93,680)(94,679)(95,678)(96,677)(97,676)(98,675)(99,674)(100,673)(101,672)(102,671)(103,670)(104,669)(105,668)(106,667)(107,666)(108,665)(109,664)(110,663)(111,662)(112,661)(113,660)(114,659)(115,658)(116,657)(117,656)(118,655)(119,654)(120,653)(121,652)(122,651)(123,650)(124,649)(125,648)(126,647)(127,646)(128,645)(129,644)(130,643)(131,642)(132,641)(133,640)(134,639)(135,638)(136,637)(137,636)(138,635)(139,634)(140,633)(141,632)(142,631)(143,630)(144,629)(145,628)(146,627)(147,626)(148,625)(149,624)(150,623)(151,622)(152,621)(153,620)(154,619)(155,618)(156,617)(157,616)(158,615)(159,614)(160,613)(161,612)(162,611)(163,610)(164,609)(165,608)(166,607)(167,606)(168,605)(169,604)(170,603)(171,602)(172,601)(173,600)(174,599)(175,598)(176,597)(177,596)(178,595)(179,594)(180,593)(181,592)(182,591)(183,590)(184,589)(185,588)(186,587)(187,586)(188,585)(189,584)(190,583)(191,582)(192,581)(193,580)(194,579)(195,578)(196,577)(197,576)(198,575)(199,574)(200,573)(201,572)(202,571)(203,570)(204,569)(205,568)(206,567)(207,566)(208,565)(209,564)(210,563)(211,562)(212,561)(213,560)(214,559)(215,558)(216,557)(217,556)(218,555)(219,554)(220,553)(221,552)(222,551)(223,550)(224,549)(225,548)(226,547)(227,546)(228,545)(229,544)(230,543)(231,542)(232,541)(233,540)(234,539)(235,538)(236,537)(237,536)(238,535)(239,534)(240,533)(241,532)(242,531)(243,530)(244,529)(245,528)(246,527)(247,526)(248,525)(249,524)(250,523)(251,522)(252,521)(253,520)(254,519)(255,518)(256,517)(257,516)(258,515)(259,514)(260,513)(261,512)(262,511)(263,510)(264,509)(265,508)(266,507)(267,506)(268,505)(269,504)(270,503)(271,502)(272,501)(273,500)(274,499)(275,498)(276,497)(277,496)(278,495)(279,494)(280,493)(281,492)(282,491)(283,490)(284,489)(285,488)(286,487)(287,486)(288,485)(289,484)(290,483)(291,482)(292,481)(293,480)(294,479)(295,478)(296,477)(297,476)(298,475)(299,474)(300,473)(301,472)(302,471)(303,470)(304,469)(305,468)(306,467)(307,466)(308,465)(309,464)(310,463)(311,462)(312,461)(313,460)(314,459)(315,458)(316,457)(317,456)(318,455)(319,454)(320,453)(321,452)(322,451)(323,450)(324,449)(325,448)(326,447)(327,446)(328,445)(329,444)(330,443)(331,442)(332,441)(333,440)(334,439)(335,438)(336,437)(337,436)(338,435)(339,434)(340,433)(341,432)(342,431)(343,430)(344,429)(345,428)(346,427)(347,426)(348,425)(349,424)(350,423)(351,422)(352,421)(353,420)(354,419)(355,418)(356,417)(357,416)(358,415)(359,414)(360,413)(361,412)(362,411)(363,410)(364,409)(365,408)(366,407)(367,406)(368,405)(369,404)(370,403)(371,402)(372,401)(373,400)(374,399)(375,398)(376,397)(377,396)(378,395)(379,394)(380,393)(381,392)(382,391)(383,390)(384,389)(385,388)(386,387)

Polycyclic presentation:

2,3,7,751|b^a=b^2,c^a=c^6,d^a=d^750

Conjugacy classes & character table

7887 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₇₇₁
maximal normal subgroup of	C₂×D₁₅₇₇₁
maximal quotient of	Dic₁₅₇₇₁, C₂×D₁₅₇₇₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31542, 12);

D15771 #44538724