Finite Group DatabasebetaFinite Group Databasebeta| order | 24870 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊(D829⋊C3) |
| unambiguous semidirect product | (C829⋊C3)⋊D5 |
| unambiguous semidirect product | D4145⋊C3 |
| Fitting, socle, derived, Hall | C5×C829 ⋊ C2×C3 |
| Hall | C5×C829⋊C3 ⋊ C2 |
| Hall | D4145 ⋊ C3 |
| Hall | C829⋊C3 ⋊ D5 |
| Sylow | C829 ⋊ C3×D5 |
| Sylow | C5 ⋊ D829⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C829 | C2×C3 |
| LCS | G ▷ C5×C829 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 834 |
| transitive degree | μt(G) = 4145 |
A minimal faithful permutation representation — 2 generators on 834 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)(338,469,470,401,402,541)(340,467,468,404,400,461)(343,394,466,406,407,462)(350,482,483,413,414,457)(357,489,490,420,421,452)(364,496,497,427,428,447)(370,503,504,434,435,442)(375,510,508,439,440,437)(380,515,501,444,445,431)(385,520,494,449,450,424)(389,525,487,454,455,417)(392,530,480,458,459,410)(403,546,547,472,473,624)(405,544,545,475,471,538)(408,465,543,477,478,539)(415,559,560,484,485,534)(422,566,567,491,492,529)(429,573,574,498,499,524)(436,580,581,505,506,519)(441,587,588,511,512,514)(446,593,585,516,517,509)(451,598,578,521,522,502)(456,603,571,526,527,495)(460,608,564,531,532,488)(463,613,557,535,536,481)(474,629,630,549,550,713)(476,627,628,552,548,621)(479,542,626,554,555,622)(486,642,643,561,562,617)(493,649,650,568,569,612)(500,656,657,575,576,607)(507,663,664,582,583,602)(513,670,671,589,590,597)(518,677,675,594,595,592)(523,682,668,599,600,586)(528,687,661,604,605,579)(533,692,654,609,610,572)(537,697,647,614,615,565)(540,702,640,618,619,558)(551,718,719,632,633,793)(553,716,717,635,631,710)(556,625,715,637,638,711)(563,730,722,644,645,706)(570,736,737,651,652,701)(577,743,744,658,659,696)(584,749,750,665,666,691)(591,755,756,672,673,686)(596,762,763,678,679,681)(601,768,760,683,684,676)(606,773,753,688,689,669)(611,778,747,693,694,662)(616,782,741,698,699,655)(620,785,734,703,704,648)(623,787,728,707,708,641)(634,738,739,721,653,807)(636,795,796,723,720,791)(639,714,794,725,726,792)(646,801,798,731,732,789)(660,812,727,745,746,784)(667,815,800,751,752,781)(674,818,806,757,758,777)(680,820,811,764,765,772)(685,822,814,769,770,767)(690,824,817,774,775,761)(695,826,816,779,780,754)(700,825,813,776,783,748)(705,823,810,771,786,742)(709,821,805,766,788,735)(712,819,799,759,790,729)(724,808,809,797,740,828)(733,829,804,802,803,827)(831,832)(833,834);(1,2,5,11,19,32,51,74,103,138,179,226,279,338,403,474,551,634,722,628,470,543,613,608,603,598,593,587,580,573,566,559,544,469,542,623,709,764,678,594,516,444,378,318,263,183,102,135,149,155,160,164,167,169,213,262,316,376,442,514,592,676,761,816,753,668,585,508,433,361,295,235,173,99,130,158,153,146,131,98,129,168,212,261,315,375,441,513,591,674,759,638,473,548,562,569,576,583,590,595,600,605,610,615,619,622,624,710,789,797,723,635,552,475,404,339,280,227,180,139,104,75,52,33,20,4,8,13,23,37,56,79,108,143,184,231,284,343,408,479,556,639,727,737,643,545,399,466,530,525,520,515,510,503,496,489,482,467,398,465,540,620,705,769,683,599,521,449,383,322,230,137,176,190,197,202,207,211,214,216,266,321,381,447,519,597,681,767,813,747,661,578,501,430,365,302,242,188,132,70,95,117,111,96,69,94,127,165,208,256,310,370,436,507,584,667,726,550,631,645,652,659,666,673,679,684,689,694,699,704,708,711,713,791,827,802,731,644,561,484,413,348,289,236,189,148,113,83,58,36,10,16,24,38,57,81,110,145,186,233,286,345,410,481,558,641,729,800,744,650,560,468,334,395,453,448,443,438,432,425,418,411,396,333,394,463,537,616,700,774,688,604,526,454,387,283,178,223,237,244,250,255,260,264,267,269,325,386,452,524,602,686,772,810,741,654,571,494,423,358,299,246,195,147,97,47,66,82,67,46,65,92,124,161,203,251,305,364,429,500,577,660,633,720,732,739,746,752,758,765,770,775,780,783,786,788,790,792,793,828,829,808,738,651,568,491,420,355,296,243,196,154,118,87,55,18,29,39,41,61,86,116,152,193,240,293,352,417,488,565,648,735,806,750,657,567,483,397,275,330,382,377,372,367,360,353,346,331,274,329,392,460,533,611,695,779,693,609,531,458,342,225,276,290,297,304,309,314,319,323,326,328,390,457,529,607,691,777,805,734,647,564,487,416,351,292,239,192,151,112,68,28,43,44,27,42,63,89,120,156,198,245,298,357,422,493,570,653,740,803,809,719,794,819,821,823,825,826,824,822,820,818,815,812,807,801,795,718,745,658,575,498,427,362,303,249,201,159,122,78,31,48,59,62,64,90,121,157,199,247,300,359,424,495,572,655,742,811,756,664,574,490,412,332,222,271,317,312,307,301,294,287,272,221,270,327,389,456,528,606,690,776,698,614,535,407,278,335,349,356,363,369,374,379,384,388,391,393,461,534,612,696,781,799,728,640,557,480,409,344,285,232,185,144,109,80,45,15,26,14,25,40,60,85,115,150,191,238,291,350,415,486,563,646,733,804,796,630,715,787,785,782,778,773,768,762,755,749,743,736,730,716,629,714,751,665,582,505,434,368,308,254,206,163,107,50,71,84,88,91,93,125,162,204,252,306,366,431,502,579,662,748,814,763,671,581,497,419,347,273,175,218,258,253,248,241,234,219,174,217,268,324,385,451,523,601,685,771,703,618,478,337,400,414,421,428,435,440,445,450,455,459,462,464,538,617,701,784,725,637,554,477,406,341,282,229,182,141,106,77,54,35,22,7,6,12,21,34,53,76,105,140,181,228,281,340,405,476,553,636,724,798,717,547,626,702,697,692,687,682,677,670,663,656,649,642,627,546,625,712,757,672,589,511,439,373,313,259,210,142,73,100,114,119,123,126,128,166,209,257,311,371,437,509,586,669,754,817,760,675,588,504,426,354,288,220,134,171,205,200,194,187,172,133,170,215,265,320,380,446,518,596,680,766,707,555,402,471,485,492,499,506,512,517,522,527,532,536,539,541,621,706,721,632,549,472,401,336,277,224,177,136,101,72,49,30,17,9,3)(830,831,833,834,832)Polycyclic presentation:
2,3,5,829|c^a=c^4,d^a=d^828,d^b=d^125702 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24870, 12);