D₁₅₇₅₁ #44538471

order	31502
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₇₅₁
unambiguous semidirect product	C₁₉⋊D₈₂₉
unambiguous semidirect product	C₈₂₉⋊D₁₉

Splittings

Fitting, socle, derived, Hall	C₁₉×C₈₂₉ ⋊ C₂
Sylow	C₈₂₉ ⋊ D₁₉
Sylow	C₁₉ ⋊ D₈₂₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₈₂₉	C₂
LCS	G ▷ C₁₉×C₈₂₉
UCS	C₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 848
transitive degree	μ_t(G) = 15751

A minimal faithful permutation representation — 2 generators on 848 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848);(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,848)(22,847)(23,846)(24,845)(25,844)(26,843)(27,842)(28,841)(29,840)(30,839)(31,838)(32,837)(33,836)(34,835)(35,834)(36,833)(37,832)(38,831)(39,830)(40,829)(41,828)(42,827)(43,826)(44,825)(45,824)(46,823)(47,822)(48,821)(49,820)(50,819)(51,818)(52,817)(53,816)(54,815)(55,814)(56,813)(57,812)(58,811)(59,810)(60,809)(61,808)(62,807)(63,806)(64,805)(65,804)(66,803)(67,802)(68,801)(69,800)(70,799)(71,798)(72,797)(73,796)(74,795)(75,794)(76,793)(77,792)(78,791)(79,790)(80,789)(81,788)(82,787)(83,786)(84,785)(85,784)(86,783)(87,782)(88,781)(89,780)(90,779)(91,778)(92,777)(93,776)(94,775)(95,774)(96,773)(97,772)(98,771)(99,770)(100,769)(101,768)(102,767)(103,766)(104,765)(105,764)(106,763)(107,762)(108,761)(109,760)(110,759)(111,758)(112,757)(113,756)(114,755)(115,754)(116,753)(117,752)(118,751)(119,750)(120,749)(121,748)(122,747)(123,746)(124,745)(125,744)(126,743)(127,742)(128,741)(129,740)(130,739)(131,738)(132,737)(133,736)(134,735)(135,734)(136,733)(137,732)(138,731)(139,730)(140,729)(141,728)(142,727)(143,726)(144,725)(145,724)(146,723)(147,722)(148,721)(149,720)(150,719)(151,718)(152,717)(153,716)(154,715)(155,714)(156,713)(157,712)(158,711)(159,710)(160,709)(161,708)(162,707)(163,706)(164,705)(165,704)(166,703)(167,702)(168,701)(169,700)(170,699)(171,698)(172,697)(173,696)(174,695)(175,694)(176,693)(177,692)(178,691)(179,690)(180,689)(181,688)(182,687)(183,686)(184,685)(185,684)(186,683)(187,682)(188,681)(189,680)(190,679)(191,678)(192,677)(193,676)(194,675)(195,674)(196,673)(197,672)(198,671)(199,670)(200,669)(201,668)(202,667)(203,666)(204,665)(205,664)(206,663)(207,662)(208,661)(209,660)(210,659)(211,658)(212,657)(213,656)(214,655)(215,654)(216,653)(217,652)(218,651)(219,650)(220,649)(221,648)(222,647)(223,646)(224,645)(225,644)(226,643)(227,642)(228,641)(229,640)(230,639)(231,638)(232,637)(233,636)(234,635)(235,634)(236,633)(237,632)(238,631)(239,630)(240,629)(241,628)(242,627)(243,626)(244,625)(245,624)(246,623)(247,622)(248,621)(249,620)(250,619)(251,618)(252,617)(253,616)(254,615)(255,614)(256,613)(257,612)(258,611)(259,610)(260,609)(261,608)(262,607)(263,606)(264,605)(265,604)(266,603)(267,602)(268,601)(269,600)(270,599)(271,598)(272,597)(273,596)(274,595)(275,594)(276,593)(277,592)(278,591)(279,590)(280,589)(281,588)(282,587)(283,586)(284,585)(285,584)(286,583)(287,582)(288,581)(289,580)(290,579)(291,578)(292,577)(293,576)(294,575)(295,574)(296,573)(297,572)(298,571)(299,570)(300,569)(301,568)(302,567)(303,566)(304,565)(305,564)(306,563)(307,562)(308,561)(309,560)(310,559)(311,558)(312,557)(313,556)(314,555)(315,554)(316,553)(317,552)(318,551)(319,550)(320,549)(321,548)(322,547)(323,546)(324,545)(325,544)(326,543)(327,542)(328,541)(329,540)(330,539)(331,538)(332,537)(333,536)(334,535)(335,534)(336,533)(337,532)(338,531)(339,530)(340,529)(341,528)(342,527)(343,526)(344,525)(345,524)(346,523)(347,522)(348,521)(349,520)(350,519)(351,518)(352,517)(353,516)(354,515)(355,514)(356,513)(357,512)(358,511)(359,510)(360,509)(361,508)(362,507)(363,506)(364,505)(365,504)(366,503)(367,502)(368,501)(369,500)(370,499)(371,498)(372,497)(373,496)(374,495)(375,494)(376,493)(377,492)(378,491)(379,490)(380,489)(381,488)(382,487)(383,486)(384,485)(385,484)(386,483)(387,482)(388,481)(389,480)(390,479)(391,478)(392,477)(393,476)(394,475)(395,474)(396,473)(397,472)(398,471)(399,470)(400,469)(401,468)(402,467)(403,466)(404,465)(405,464)(406,463)(407,462)(408,461)(409,460)(410,459)(411,458)(412,457)(413,456)(414,455)(415,454)(416,453)(417,452)(418,451)(419,450)(420,449)(421,448)(422,447)(423,446)(424,445)(425,444)(426,443)(427,442)(428,441)(429,440)(430,439)(431,438)(432,437)(433,436)(434,435)

Polycyclic presentation:

2,19,829|b^a=b^18,c^a=c^828

Conjugacy classes & character table

7877 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₇₅₁
maximal normal subgroup of	C₂×D₁₅₇₅₁
maximal quotient of	Dic₁₅₇₅₁, C₂×D₁₅₇₅₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31502, 3);

D15751 #44538471