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Aliases
Splittings
Subgroups
8 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 828 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,145)(98,122,142)(101,125,139)(104,128,136)(107,131,133)(110,134,130)(113,137,127)(116,140,124)(118,143,121)(120,147,176)(123,150,173)(126,153,170)(129,156,167)(132,159,164)(135,162,161)(138,165,158)(141,168,155)(144,171,152)(146,174,149)(148,178,210)(151,181,207)(154,184,204)(157,187,201)(160,190,198)(163,193,195)(166,196,192)(169,199,189)(172,202,186)(175,205,183)(177,208,180)(179,212,247)(182,215,244)(185,218,241)(188,221,238)(191,224,235)(194,227,232)(197,230,229)(200,233,226)(203,236,223)(206,239,220)(209,242,217)(211,245,214)(213,249,287)(216,252,284)(219,255,281)(222,258,278)(225,261,275)(228,264,272)(231,267,269)(234,270,266)(237,273,263)(240,276,260)(243,279,257)(246,282,254)(248,285,251)(250,289,330)(253,292,327)(256,295,324)(259,298,321)(262,301,318)(265,304,315)(268,307,312)(271,310,309)(274,313,306)(277,316,303)(280,319,300)(283,322,297)(286,325,294)(288,328,291)(290,332,376)(293,335,373)(296,338,370)(299,341,367)(302,344,364)(305,347,361)(308,350,358)(311,353,355)(314,356,352)(317,359,349)(320,362,346)(323,365,343)(326,368,340)(329,371,337)(331,374,334)(333,378,425)(336,381,422)(339,384,419)(342,387,416)(345,390,413)(348,393,410)(351,396,407)(354,399,404)(357,402,401)(360,405,398)(363,408,395)(366,411,392)(369,414,389)(372,417,386)(375,420,383)(377,423,380)(379,427,433)(382,429,473)(385,432,470)(388,435,467)(391,438,464)(394,441,461)(397,444,458)(400,447,455)(403,450,452)(406,453,449)(409,456,446)(412,459,443)(415,462,440)(418,465,437)(421,468,434)(424,471,431)(426,474,428)(430,478,480)(436,482,517)(439,485,514)(442,488,511)(445,491,508)(448,494,505)(451,497,502)(454,500,499)(457,503,496)(460,506,493)(463,509,490)(466,512,487)(469,515,484)(472,518,481)(475,520,479)(476,483,477)(486,526,558)(489,529,555)(492,532,552)(495,535,549)(498,538,546)(501,541,543)(504,544,540)(507,547,537)(510,550,534)(513,553,531)(516,556,528)(519,559,525)(521,561,524)(522,527,523)(530,567,596)(533,570,593)(536,573,590)(539,576,587)(542,579,584)(545,582,581)(548,585,578)(551,588,575)(554,591,572)(557,594,569)(560,597,566)(562,599,565)(563,568,564)(571,605,631)(574,608,628)(577,611,625)(580,614,622)(583,617,619)(586,620,616)(589,623,613)(592,626,610)(595,629,607)(598,632,604)(600,634,603)(601,606,602)(609,640,663)(612,643,660)(615,646,657)(618,649,654)(621,652,651)(624,655,648)(627,658,645)(630,661,642)(633,664,639)(635,666,638)(636,641,637)(644,672,692)(647,675,689)(650,678,686)(653,681,683)(656,684,680)(659,687,677)(662,690,674)(665,693,671)(667,695,670)(668,673,669)(676,701,718)(679,704,715)(682,707,712)(685,710,709)(688,713,706)(691,716,703)(694,719,700)(696,721,699)(697,702,698)(705,727,741)(708,730,738)(711,733,735)(714,736,732)(717,739,729)(720,742,726)(722,744,725)(723,728,724)(731,750,761)(734,753,758)(737,756,755)(740,759,752)(743,762,749)(745,764,748)(746,751,747)(754,770,778)(757,773,775)(760,776,772)(763,779,769)(765,781,768)(766,771,767)(774,787,792)(777,790,789)(780,793,786)(782,795,785)(783,788,784)(791,801,803)(794,804,800)(796,806,799)(797,802,798)(805,812,811)(807,814,810)(808,813,809)(815,819,818)(816,820,817)(821,823,822)(824,825,826,827,828);(1,2,4,8,15,25,38,54,73,95,120,148,179,213,250,290,333,379,381,289,331,375,421,469,516,557,595,630,662,691,717,740,760,777,791,802,781,742,690,626,550,459,362,276,202,140,90,51,67,86,108,133,161,192,226,263,303,346,392,440,487,528,566,603,637,605,529,438,341,255,181,119,146,175,206,240,277,317,360,406,454,501,542,580,615,647,676,702,666,597,515,417,325,245,176,207,241,278,318,361,407,455,502,543,581,616,648,677,703,726,748,767,750,704,646,576,494,396,304,224,156,100,56,24,36,50,66,85,107,132,160,191,225,262,302,345,391,439,486,527,474,376,422,470,517,558,596,631,663,692,718,741,761,778,792,803,811,818,822,819,804,776,736,684,620,544,453,356,270,196,134,84,46,20,5,9,16,26,39,55,74,96,121,149,180,214,251,291,334,380,428,477,432,335,249,288,329,372,418,466,513,554,592,627,659,688,714,737,757,774,788,764,719,661,591,509,411,319,239,171,115,70,89,111,136,164,195,229,266,306,349,395,443,490,531,569,604,638,669,640,570,488,390,298,218,150,94,118,144,172,203,237,274,314,357,403,451,498,539,577,612,644,673,634,559,468,371,285,210,244,281,321,364,410,458,505,546,584,619,651,680,706,729,749,768,784,770,730,678,614,538,447,350,264,190,128,78,40,14,23,34,47,63,82,104,129,157,188,222,259,299,342,388,436,483,425,473,427,476,522,563,601,636,668,697,723,746,766,783,797,808,816,821,823,814,793,759,713,655,585,503,405,313,233,165,109,65,33,12,19,29,42,58,77,99,124,152,183,217,254,294,337,383,431,479,523,482,384,292,212,248,286,326,369,415,463,510,551,589,624,656,685,711,734,754,771,744,693,629,553,462,365,279,205,143,92,114,139,167,198,232,269,309,352,398,446,493,534,572,607,639,670,698,672,608,532,441,344,258,184,122,72,93,116,141,169,200,234,271,311,354,400,448,495,536,574,609,641,599,518,420,328,247,284,324,367,413,461,508,549,587,622,654,683,709,732,752,769,785,798,787,753,707,649,579,497,399,307,227,159,103,59,27,7,13,21,31,44,60,79,101,126,154,185,219,256,296,339,385,433,480,478,378,426,475,521,562,600,635,667,696,722,745,765,782,796,807,815,820,806,779,739,687,623,547,456,359,273,199,137,87,49,22,32,45,61,80,102,127,155,186,220,257,297,340,386,434,481,524,564,526,435,338,252,178,211,246,283,323,366,412,460,507,548,586,621,653,682,708,731,751,721,664,594,512,414,322,242,174,117,142,170,201,235,272,312,355,401,449,496,537,575,610,642,671,699,724,701,643,573,491,393,301,221,153,97,53,71,91,113,138,166,197,231,268,308,351,397,445,492,533,571,606,561,471,374,287,327,370,416,464,511,552,590,625,657,686,712,735,755,772,786,799,809,801,773,733,681,617,541,450,353,267,193,131,81,43,17,3,6,11,18,28,41,57,76,98,123,151,182,216,253,293,336,382,430,429,332,377,424,472,519,560,598,633,665,694,720,743,763,780,794,805,813,795,762,716,658,588,506,408,316,236,168,112,68,35,48,64,83,105,130,158,189,223,260,300,343,389,437,484,525,565,602,567,485,387,295,215,147,177,209,243,280,320,363,409,457,504,545,583,618,650,679,705,728,695,632,556,465,368,282,208,145,173,204,238,275,315,358,404,452,499,540,578,613,645,674,700,725,747,727,675,611,535,444,347,261,187,125,75,37,52,69,88,110,135,163,194,228,265,305,348,394,442,489,530,568,520,423,330,373,419,467,514,555,593,628,660,689,715,738,758,775,789,800,810,817,812,790,756,710,652,582,500,402,310,230,162,106,62,30,10)
Conjugacy classes & character table
1385 conjugacy classes. The character table is too large to display.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12345, 1);
Finite Group Databasebeta