Finite Group Databasebeta

?

← Prev (#36384239) Next (#36384241) →

C₅⋊(D₇₅₇⋊C₃) #36384240

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Splittings

Subgroups

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 762 points:

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

Polycyclic presentation:

2,3,5,757|c^a=c^4,d^a=d^756,d^b=d^27

Conjugacy classes & character table

642 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22710, 12);