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Aliases
Splittings
Subgroups
22 conjugacy classes of subgroups.
Full subgroup data →
Presentation
A minimal faithful permutation representation — 2 generators on 771 points:
(2,3,4,7,5,8,10,16)(6,12,13,28,14,29,31,63)(9,19,20,45,21,46,48,37)(11,24,25,56,26,57,23,54)(15,33,34,75,35,76,78,157)(17,38,39,85,40,18,42,43)(22,50,51,107,52,108,110,132)(27,59,60,125,61,126,70,148)(30,66,67,139,68,140,142,84)(32,71,72,150,73,151,55,117)(36,80,81,168,82,169,171,306)(41,87,88,183,89,65,136,137)(44,93,94,64,95,133,194,134)(47,98,99,202,100,203,102,116)(49,103,104,210,105,120,91,189)(53,112,113,224,114,225,227,293)(58,121,122,191,123,97,200,92)(62,128,129,252,130,253,163,300)(69,144,145,274,146,275,223,259)(74,153,154,286,155,287,149,279)(77,160,161,297,162,298,167,177)(79,164,165,302,166,303,118,235)(83,173,174,314,175,315,317,512)(86,179,180,119,181,237,178,238)(90,185,186,335,187,159,294,295)(96,196,197,158,198,236,349,172)(101,205,206,363,207,364,219,278)(106,211,147,277,212,247,263,420)(109,215,216,378,217,379,271,233)(111,220,221,385,222,281,190,342)(115,229,230,138,231,265,135,261)(124,242,243,411,244,214,376,264)(127,248,249,343,250,266,421,192)(131,255,256,433,257,434,313,510)(141,267,268,398,269,449,209,234)(143,272,273,455,228,240,193,345)(152,282,283,347,284,201,356,195)(156,289,290,475,291,476,301,492)(170,309,310,504,311,505,507,323)(176,319,320,521,321,522,524,711)(182,326,327,246,328,417,346,418)(184,331,332,280,333,464,324,465)(188,337,338,548,339,308,497,502)(199,351,352,307,353,499,501,500)(204,359,360,553,361,358,355,341)(208,366,367,578,368,579,391,491)(213,372,373,585,374,428,482,642)(218,381,382,596,383,597,386,462)(226,388,389,603,390,604,305,397)(232,393,394,466,395,610,538,678)(239,403,404,330,405,260,440,441)(241,407,408,371,409,402,325,531)(245,413,414,630,415,387,601,483)(251,423,424,640,425,456,595,446)(254,429,430,556,431,486,490,344)(258,436,437,650,438,651,518,619)(262,443,369,581,444,296,484,485)(270,451,432,643,452,665,384,463)(276,457,458,615,459,629,454,399)(285,468,416,632,469,377,590,447)(288,312,472,509,473,448,661,348)(292,478,479,695,480,611,511,682)(299,487,488,617,489,547,370,400)(304,494,495,508,496,357,568,350)(316,445,515,657,516,658,503,529)(318,406,519,622,520,623,401,618)(322,525,526,714,527,666,649,766)(329,534,535,427,536,583,561,506)(334,540,541,471,542,687,559,688)(336,544,545,493,546,699,530,700)(340,549,550,727,551,514,708,565)(354,562,563,513,564,707,732,704)(362,570,426,641,571,573,589,636)(365,574,575,655,576,591,637,554)(375,586,587,728,588,645,703,552)(380,592,593,729,594,662,566,555)(392,607,442,481,608,419,635,614)(396,612,613,689,560,477,609,694)(410,498,616,461,624,634,656,675)(412,626,627,599,628,677,532,715)(422,638,639,670,605,620,533,716)(435,646,647,582,648,706,701,557)(439,652,653,697,654,580,713,740)(450,663,664,528,631,569,567,558)(453,667,668,765,669,523,606,698)(460,672,673,753,674,761,600,676)(467,679,680,584,681,621,539,720)(470,683,684,767,685,602,751,696)(474,517,686,710,691,671,748,660)(537,717,718,644,719,726,705,702)(543,723,724,693,725,722,731,690)(572,734,735,769,736,739,750,764)(577,737,692,712,738,744,743,762)(598,745,746,768,747,757,763,730)(625,755,756,742,709,741,754,659)(633,758,759,749,760,752,721,733);(1,2,6,15,36,83,176,322,528,455,670,724,535,197,350,507,635,403,93,193,221,104,39,86,182,329,537,618,754,768,729,385,599,749,727,564,353,198,95,38,3,9,22,53,115,232,396,509,347,194,346,559,617,302,493,701,650,519,196,165,72,25,4,11,27,62,131,258,439,655,553,210,371,584,718,352,500,306,261,87,12,30,69,147,88,184,334,543,679,440,656,615,398,150,280,466,314,513,351,494,282,121,19,47,101,208,369,582,556,343,191,85,178,324,530,495,702,510,443,185,33,77,154,60,13,32,74,156,292,481,603,378,202,56,119,239,406,563,562,502,468,242,50,109,218,373,186,336,547,638,407,179,325,532,664,283,467,682,515,337,80,170,312,248,66,141,270,453,657,504,297,139,45,7,17,41,90,188,340,552,683,413,112,226,290,129,34,79,167,305,498,326,533,593,360,122,241,410,625,549,173,316,517,423,144,276,460,587,338,496,303,272,103,24,58,124,245,416,633,761,710,632,229,392,609,429,160,299,490,697,755,534,488,268,99,20,49,106,213,375,319,523,712,641,277,461,479,256,81,172,157,293,211,59,127,251,426,243,412,629,757,758,393,611,753,643,286,471,689,521,550,717,699,663,359,98,204,362,572,684,525,665,640,411,183,330,538,646,309,506,300,491,372,128,254,432,206,51,111,223,386,600,612,476,596,363,125,246,419,437,174,318,295,483,570,205,365,577,735,414,631,449,662,626,331,539,721,766,472,690,492,698,586,255,435,649,574,215,380,595,743,691,473,287,274,107,28,64,135,262,445,659,642,764,734,366,580,668,367,113,228,142,271,454,540,722,746,575,249,422,604,653,320,436,485,696,737,381,598,748,685,469,284,151,281,220,71,152,285,470,686,747,597,744,767,581,651,524,713,388,605,421,637,763,723,687,458,216,67,143,227,391,606,652,579,739,769,585,742,658,444,230,134,63,132,259,153,288,474,692,424,594,379,591,526,648,434,645,765,475,693,661,714,759,680,464,677,592,267,450,601,750,738,576,364,573,630,335,520,317,518,607,418,148,278,462,289,477,673,382,145,222,110,219,384,431,253,428,578,252,427,505,706,394,441,137,264,446,451,666,751,736,571,361,203,358,569,544,726,708,711,694,688,279,463,676,478,610,752,745,457,628,376,589,425,250,126,247,224,75,158,171,313,511,675,420,636,762,667,522,588,374,212,105,48,102,209,370,583,741,740,430,489,298,486,613,608,265,447,660,672,760,590,415,244,123,57,120,240,164,304,497,703,674,459,275,456,671,516,315,514,709,624,409,200,355,566,639,417,634,389,161,166,78,163,301,390,225,387,602,728,551,339,187,89,40,16,37,84,177,323,529,669,452,269,140,266,448,311,169,308,503,695,681,356,567,627,408,237,402,620,487,546,294,482,383,217,108,214,377,548,707,732,623,405,238,54,116,233,397,614,480,291,155,73,31,70,149,162,76,159,296,433,644,568,545,332,180,43,92,192,344,557,484,368,207,100,46,97,201,357,499,704,512,678,465,117,234,399,616,404,621,725,542,333,136,263,146,68,29,65,138,168,307,501,705,720,531,189,341,554,654,438,257,130,61,26,10,23,55,118,236,401,619,647,700,235,400,541,327,94,195,348,560,395,231,114,52,21,8,18,44,96,199,354,565,733,715,342,555,730,756,622,719,536,328,181,42,91,190,273,133,260,442,310,508,349,561,731,716,345,558,527,321,175,82,35,14,5)(770,771)
Conjugacy classes & character table
208 conjugacy classes — view the full character table.
Relationships
GAP
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12304, 53);
Finite Group Databasebeta