Finite Group DatabasebetaFinite Group Databasebeta| order | 20181 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Hall | C7×C961 ⋊ C3 |
| Sylow | C961 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C961⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C961 | C3 |
| Frattini | C31 | #10620253 |
| socle | C7×C31 | C31⋊C3 |
| LCS | G ▷ C7×C961 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 968 |
| transitive degree | μt(G) = 6727 |
A minimal faithful permutation representation — 2 generators on 968 points:
(2,4,9)(3,7,17)(5,12,29)(6,15,37)(8,20,49)(10,24,22)(11,27,42)(13,32,74)(14,35,81)(16,40,93)(18,44,23)(19,47,43)(21,52,116)(25,58,67)(26,61,86)(28,65,109)(30,69,54)(31,72,98)(33,77,160)(34,79,145)(36,84,175)(38,88,55)(39,91,99)(41,96,198)(45,103,68)(46,78,87)(48,53,110)(50,111,56)(51,114,100)(57,63,101)(59,125,153)(60,128,168)(62,131,191)(64,107,102)(66,138,230)(70,144,140)(71,147,180)(73,151,219)(75,155,119)(76,158,202)(80,166,307)(82,170,120)(83,173,203)(85,178,328)(89,184,141)(90,186,181)(92,189,220)(94,193,121)(95,196,204)(97,201,135)(104,211,154)(105,214,169)(106,139,192)(108,218,231)(112,223,142)(113,226,182)(115,229,221)(117,232,122)(118,235,205)(123,136,222)(124,149,206)(126,244,295)(127,245,269)(129,248,321)(130,188,207)(132,252,356)(133,209,143)(134,163,183)(137,228,208)(146,271,312)(148,274,346)(150,217,210)(152,281,400)(156,286,263)(157,289,332)(159,293,385)(161,297,237)(162,300,365)(164,287,238)(165,305,366)(167,310,500)(171,315,264)(172,316,333)(174,319,386)(176,323,239)(177,326,367)(179,331,278)(185,338,283)(187,340,347)(190,344,401)(194,350,265)(195,352,334)(197,355,387)(199,358,240)(200,361,368)(212,378,296)(213,379,270)(215,282,322)(216,345,357)(224,391,284)(225,394,313)(227,397,348)(233,404,266)(234,407,335)(236,410,388)(241,261,402)(242,279,389)(243,291,369)(246,417,493)(247,318,370)(249,421,522)(250,343,363)(251,354,371)(253,349,256)(254,376,285)(255,381,314)(257,383,403)(258,390,267)(259,396,336)(260,399,364)(262,409,372)(268,276,373)(272,444,514)(273,342,374)(275,448,555)(277,302,337)(280,398,375)(288,464,504)(290,467,544)(292,384,377)(294,473,602)(298,478,436)(299,481,531)(301,485,586)(303,479,437)(304,488,532)(306,491,587)(308,495,411)(309,498,567)(311,503,470)(317,508,545)(320,512,603)(324,517,438)(325,518,533)(327,521,588)(329,524,412)(330,527,568)(339,538,475)(341,539,556)(351,550,459)(353,552,546)(359,558,439)(360,560,534)(362,563,589)(380,474,494)(382,513,523)(392,593,476)(393,594,442)(395,597,515)(405,608,460)(406,611,505)(408,614,547)(413,457,604)(414,471,590)(415,483,569)(416,490,570)(418,622,701)(419,511,564)(420,520,571)(422,549,452)(423,543,605)(424,554,565)(425,562,572)(426,579,477)(427,581,463)(428,449,516)(429,540,557)(430,592,461)(431,596,506)(432,599,548)(433,606,440)(434,613,535)(435,616,566)(441,454,591)(443,510,573)(445,642,724)(446,542,529)(447,553,574)(450,576,462)(451,583,507)(453,584,607)(455,598,536)(456,601,530)(458,615,575)(465,662,718)(466,541,577)(468,666,752)(469,487,537)(472,600,578)(480,679,710)(482,682,742)(484,585,580)(486,687,625)(489,689,743)(492,692,787)(496,696,636)(497,697,734)(499,700,774)(501,703,617)(502,706,761)(509,713,753)(519,722,744)(525,726,637)(526,727,735)(528,730,775)(551,749,674)(559,754,657)(561,756,745)(582,693,702)(595,780,694)(609,650,675)(610,794,639)(612,797,719)(618,672,788)(619,685,776)(620,691,759)(621,699,762)(623,748,668)(624,717,789)(626,729,763)(627,758,760)(628,667,695)(629,714,725)(630,778,676)(631,779,660)(632,782,720)(633,792,658)(634,796,711)(635,799,746)(638,654,790)(640,716,731)(641,723,764)(643,721,649)(644,741,791)(645,751,732)(646,757,765)(647,769,677)(648,772,678)(651,777,659)(652,781,712)(653,784,747)(655,798,736)(656,801,733)(661,715,766)(663,829,874)(664,740,708)(665,750,767)(669,773,793)(670,783,737)(671,786,709)(673,800,768)(680,843,870)(681,739,770)(683,846,814)(684,688,738)(686,785,771)(690,850,889)(698,856,883)(704,858,812)(705,859,879)(707,862,898)(728,875,884)(755,834,838)(795,910,853)(802,848,804)(803,852,905)(805,861,892)(806,877,849)(807,851,857)(808,901,854)(809,818,839)(810,909,826)(811,912,871)(813,869,906)(815,876,893)(816,891,878)(817,830,855)(819,899,840)(820,900,841)(821,902,872)(822,907,825)(823,911,866)(824,914,885)(827,868,863)(828,873,894)(831,882,908)(832,888,864)(833,890,895)(835,904,886)(836,913,880)(837,915,865)(842,867,896)(844,932,925)(845,887,897)(847,903,881)(860,937,945)(916,938,934)(917,953,935)(918,944,933)(919,929,921)(920,950,936)(922,952,930)(923,954,941)(924,940,951)(926,943,948)(927,947,939)(928,955,946)(931,942,949)(956,961,959)(957,960,958)(963,964,965)(966,968,967);(1,109,215,366,558,11,363,476,762,755,73,732,839,853,471,294,939,496,937,280,318,876,648,710,645,9,191,428,532,726,57,110,216,367,405,242,364,540,763,685,618,733,176,902,472,490,910,497,772,832,29,321,649,298,858,143,192,429,533,633,441,548,218,368,406,311,885,40,599,686,185,903,177,381,911,74,493,499,75,777,285,322,650,299,822,659,720,344,534,634,740,941,131,229,408,45,600,420,78,613,160,369,804,12,390,477,494,834,76,708,840,854,512,301,823,868,959,274,355,635,133,750,641,186,235,46,98,602,58,111,187,370,905,243,331,509,764,692,619,709,851,948,467,521,824,276,873,828,316,361,134,180,752,144,193,48,99,603,80,549,190,371,359,418,886,513,765,682,700,928,166,901,931,488,527,277,312,874,286,323,135,181,753,411,721,423,100,117,85,898,392,372,360,422,955,417,597,728,171,706,469,379,909,478,495,278,77,778,636,855,644,182,261,412,945,630,535,118,97,801,393,585,362,38,598,684,126,611,172,373,470,244,391,489,766,831,313,457,637,949,819,711,262,343,891,631,739,627,7,228,395,13,586,39,101,604,127,550,174,374,906,442,672,492,767,704,826,458,511,944,820,867,816,27,354,632,119,742,130,17,230,14,587,419,102,605,129,848,803,375,329,445,673,691,768,705,830,927,72,520,821,263,870,273,37,356,120,743,640,183,231,16,588,773,536,96,132,849,351,880,330,449,954,158,699,707,164,900,466,81,522,264,769,827,314,357,121,744,882,712,252,21,589,112,786,626,139,799,300,487,916,35,596,681,145,701,165,376,907,463,523,265,770,940,841,448,122,745,258,888,815,340,410,113,217,800,128,226,353,25,590,416,103,606,146,702,459,377,908,465,666,266,771,454,844,926,508,563,259,342,890,271,352,115,2,219,380,184,232,26,564,674,537,607,148,846,460,881,310,468,958,689,730,455,510,943,464,518,260,22,346,628,315,358,3,220,382,567,609,28,565,675,892,622,152,878,324,862,670,474,953,679,697,456,67,514,817,479,524,23,347,629,734,809,123,221,383,568,610,413,566,94,784,847,339,797,325,583,671,153,594,698,156,703,68,378,818,480,921,267,348,543,735,810,638,746,20,399,612,104,785,95,163,798,295,569,327,30,591,154,538,608,157,863,461,515,717,482,922,825,871,65,554,811,254,887,251,47,409,105,202,625,4,222,296,570,754,31,529,676,694,852,159,864,929,935,151,486,923,450,942,447,91,562,255,332,814,24,49,106,203,787,124,201,341,571,559,414,530,714,893,293,802,865,308,950,665,173,729,451,504,925,69,93,256,333,889,268,349,108,204,233,167,747,345,572,485,623,946,84,782,845,305,861,309,581,952,155,175,452,161,899,462,516,257,334,433,617,872,539,205,234,179,915,248,397,561,89,783,621,214,796,297,307,668,32,592,677,695,453,162,654,812,936,713,335,434,542,947,444,552,236,18,398,595,33,407,90,206,788,125,223,317,573,669,415,503,690,894,850,505,655,716,960,662,722,435,63,553,808,237,560,19,42,400,34,350,92,207,789,246,748,320,574,525,639,836,693,895,843,856,656,149,723,920,436,727,64,86,555,238,517,250,43,401,36,774,624,208,199,249,934,551,575,526,643,837,291,780,860,303,859,150,168,724,437,696,446,87,556,239,883,813,336,52,41,775,224,736,200,253,914,483,688,528,82,781,292,269,794,304,576,664,169,593,438,896,924,506,138,240,884,430,866,425,53,616,225,384,806,15,396,484,59,776,83,209,790,270,749,306,577,951,660,281,439,897,651,930,646,189,758,431,541,918,61,114,227,5,385,247,44,402,60,759,620,210,791,272,473,657,578,501,663,833,319,877,652,715,957,147,196,432,54,544,443,88,116,6,386,582,337,403,62,687,838,737,178,275,933,491,938,502,667,961,289,326,653,140,718,661,170,198,55,545,807,507,557,8,387,584,761,421,66,760,194,904,805,282,912,481,498,835,79,779,842,287,328,141,579,919,678,725,56,546,741,879,642,241,388,50,601,795,212,614,195,302,913,245,394,519,70,500,283,211,792,288,857,142,580,869,680,829,440,547,136,751,917,426,756,51,107,615,213,365,197,10,389,475,338,404,71,731,284,738,793,290,932,658,719,279,683,956,647,875,137,188,757,427,531,424)(962,963,965,967,964,966,968)Polycyclic presentation:
3,7,31^2|c^31=d^20,b^a=b^4,c^a=c^5d,d^a=d^52245 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20181, 5);