Finite Group Databasebeta

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C₅⋊(D₁₀₅₁⋊C₃) #44538666

• 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 1056 points:

(2,3)(4,5)(7,11,12,8,9,13)(10,19,20,14,15,29)(16,32,33,22,23,46)(17,31,27,25,21,18)(24,51,52,35,36,69)(26,49,50,38,34,43)(28,30,48,40,41,44)(37,74,75,54,55,98)(39,72,73,57,53,66)(42,47,71,59,60,67)(45,87,85,63,64,62)(56,103,104,77,78,133)(58,101,102,80,76,95)(61,70,100,82,83,96)(65,116,117,88,89,91)(68,122,114,92,93,86)(79,138,139,106,107,174)(81,136,137,109,105,130)(84,99,135,111,112,131)(90,151,152,118,119,126)(94,158,156,123,124,121)(97,163,149,127,128,115)(108,179,180,141,142,221)(110,177,178,144,140,171)(113,134,176,146,147,172)(120,192,193,153,154,167)(125,199,200,159,160,162)(129,205,197,164,165,157)(132,210,190,168,169,150)(143,226,227,182,183,274)(145,224,225,185,181,218)(148,175,223,187,188,219)(155,239,240,194,195,214)(161,246,247,201,202,209)(166,253,251,206,207,204)(170,258,244,211,212,198)(173,263,237,215,216,191)(184,279,280,229,230,333)(186,277,278,232,228,271)(189,222,276,234,235,272)(196,292,293,241,242,267)(203,299,300,248,249,262)(208,306,307,254,255,257)(213,312,304,259,260,252)(217,317,297,264,265,245)(220,322,290,268,269,238)(231,338,339,282,283,398)(233,336,337,285,281,330)(236,275,335,287,288,331)(243,351,352,294,295,326)(250,358,359,301,302,321)(256,365,366,308,309,316)(261,372,370,313,314,311)(266,377,363,318,319,305)(270,382,356,323,324,298)(273,387,349,327,328,291)(284,403,404,341,342,469)(286,401,402,344,340,395)(289,334,400,346,347,396)(296,416,417,353,354,391)(303,423,424,360,361,386)(310,430,431,367,368,381)(315,437,438,373,374,376)(320,443,435,378,379,371)(325,448,428,383,384,364)(329,453,421,388,389,357)(332,458,414,392,393,350)(343,474,475,406,407,546)(345,472,473,409,405,466)(348,399,471,411,412,467)(355,487,488,418,419,462)(362,494,495,425,426,457)(369,501,502,432,433,452)(375,508,509,439,440,447)(380,515,513,444,445,442)(385,520,506,449,450,436)(390,525,499,454,455,429)(394,530,492,459,460,422)(397,535,485,463,464,415)(408,551,552,477,478,629)(410,549,550,480,476,543)(413,470,548,482,483,544)(420,564,565,489,490,539)(427,571,572,496,497,534)(434,578,579,503,504,529)(441,585,586,510,511,524)(446,592,593,516,517,519)(451,598,590,521,522,514)(456,603,583,526,527,507)(461,608,576,531,532,500)(465,613,569,536,537,493)(468,618,562,540,541,486)(479,634,635,554,555,718)(481,632,633,557,553,626)(484,547,631,559,560,627)(491,647,648,566,567,622)(498,654,655,573,574,617)(505,661,662,580,581,612)(512,668,669,587,588,607)(518,675,676,594,595,602)(523,682,680,599,600,597)(528,687,673,604,605,591)(533,692,666,609,610,584)(538,697,659,614,615,577)(542,702,652,619,620,570)(545,707,645,623,624,563)(556,723,724,637,638,813)(558,721,722,640,636,715)(561,630,720,642,643,716)(568,736,737,649,650,711)(575,743,744,656,657,706)(582,750,751,663,664,701)(589,757,758,670,671,696)(596,764,765,677,678,691)(601,771,772,683,684,686)(606,777,769,688,689,681)(611,782,762,693,694,674)(616,787,755,698,699,667)(621,792,748,703,704,660)(625,797,741,708,709,653)(628,802,734,712,713,646)(639,818,819,726,727,906)(641,816,817,729,725,810)(644,719,815,731,732,811)(651,830,831,738,739,806)(658,837,838,745,746,801)(665,844,845,752,753,796)(672,851,852,759,760,791)(679,856,857,766,767,786)(685,863,864,773,774,781)(690,870,868,778,779,776)(695,875,861,783,784,770)(700,880,822,788,789,763)(705,885,849,793,794,756)(710,890,842,798,799,749)(714,894,835,803,804,742)(717,898,828,807,808,735)(728,886,887,821,795,963)(730,908,909,823,820,904)(733,814,907,825,826,905)(740,914,915,832,833,902)(747,920,921,839,840,897)(754,925,926,846,847,893)(761,930,931,853,854,889)(768,935,936,858,859,884)(775,940,941,865,866,879)(780,947,948,871,872,874)(785,953,945,876,877,869)(790,958,911,881,882,862)(800,967,827,891,892,850)(805,969,913,895,896,843)(809,971,919,899,900,836)(812,972,912,901,903,829)(824,964,965,910,888,975)(834,976,977,916,917,974)(841,978,979,922,923,973)(848,982,983,927,928,970)(855,987,988,932,933,968)(860,992,993,937,938,966)(867,997,998,942,943,962)(873,1002,1003,949,950,957)(878,1007,981,954,955,952)(883,1012,918,959,960,946)(924,1013,1014,980,961,1020)(929,1021,1022,984,985,1019)(934,1023,1024,989,990,1018)(939,1027,1028,994,995,1017)(944,1032,1033,999,1000,1016)(951,1037,1026,1004,1005,1015)(956,1041,986,1008,1009,1011)(991,1042,1043,1025,1010,1047)(996,1048,1049,1029,1030,1046)(1001,1050,1051,1034,1035,1045)(1006,1053,1031,1038,1039,1044)(1036,1054,1055,1052,1040,1056);(1,2,4,5,3)(6,7,10,16,24,37,56,79,108,143,184,231,284,343,408,479,556,639,728,822,762,673,590,513,438,366,300,240,178,104,135,163,158,151,136,103,134,173,217,266,320,380,446,518,596,679,768,860,939,996,1031,1033,993,931,845,744,648,550,404,471,535,530,525,520,515,508,501,494,487,472,403,470,545,625,710,800,727,820,833,840,847,854,859,866,872,877,882,887,892,896,900,903,905,906,975,1012,1007,1002,997,992,987,982,978,976,964,886,793,703,619,540,412,283,340,354,361,368,374,379,384,389,393,396,398,466,539,617,701,791,884,962,1015,1025,989,932,853,759,670,587,510,439,373,313,259,211,168,112,55,76,89,93,96,98,130,167,209,257,311,371,436,507,584,667,756,850,825,731,642,559,482,411,346,287,234,187,146,111,82,59,40,27,12,11,17,26,39,58,81,110,145,186,233,286,345,410,481,558,641,730,824,911,861,769,680,593,509,431,359,293,225,139,176,210,205,199,192,177,138,175,220,270,325,385,451,523,601,685,775,867,944,1001,1036,1051,1028,988,926,838,737,633,475,548,618,613,608,603,598,592,585,578,571,564,549,474,547,628,714,805,826,638,725,739,746,753,760,767,774,779,784,789,794,799,804,808,811,813,904,974,1020,1041,1037,1032,1027,1023,1021,1013,959,881,788,698,614,536,463,347,230,281,295,302,309,314,319,324,328,331,333,395,462,534,612,696,786,879,957,1011,984,927,846,752,663,580,503,432,367,308,254,206,164,127,83,36,53,64,67,69,95,126,162,204,252,305,364,429,500,577,660,749,843,912,828,734,645,562,485,414,349,290,237,190,149,114,85,50,20,31,19,30,45,65,90,120,155,196,243,296,355,420,491,568,651,740,834,918,945,868,772,676,586,502,424,352,278,180,223,263,258,253,246,239,224,179,222,273,329,390,456,528,606,690,780,873,951,1006,1040,1055,1049,1024,983,921,831,722,552,631,707,702,697,692,687,682,675,668,661,654,647,632,551,630,717,809,901,732,555,636,650,657,664,671,678,684,689,694,699,704,709,713,716,718,810,902,973,1019,1047,1053,1050,1048,1042,1008,954,876,783,693,609,531,459,392,288,183,228,242,249,255,260,265,269,272,274,330,391,457,529,607,691,781,874,952,980,922,839,745,656,573,496,425,360,301,248,201,159,123,92,60,23,34,44,46,66,91,121,157,198,245,298,357,422,493,570,653,742,836,919,835,741,652,569,492,421,356,297,244,197,156,117,73,33,48,49,32,47,68,94,125,161,203,250,303,362,427,498,575,658,747,841,924,981,948,864,765,669,579,495,417,337,227,276,322,317,312,306,299,292,277,226,275,332,394,461,533,611,695,785,878,956,1010,1030,1035,1039,1043,1022,979,915,817,635,720,802,797,792,787,782,777,771,764,757,750,743,736,721,634,719,812,899,807,643,478,553,567,574,581,588,595,600,605,610,615,620,624,627,629,715,806,897,970,1018,1046,1056,1054,1038,1004,949,871,778,688,604,526,454,388,327,235,142,181,195,202,207,212,216,219,221,271,326,386,452,524,602,686,776,869,946,916,832,738,649,566,489,418,353,294,241,194,153,118,88,63,41,15,21,29,43,62,86,115,150,191,238,291,350,415,486,563,646,735,829,913,842,748,659,576,499,428,363,304,251,200,152,102,52,71,87,72,51,70,97,129,166,208,256,310,369,434,505,582,665,754,848,929,986,1003,941,857,758,662,572,488,402,280,335,387,382,377,372,365,358,351,336,279,334,397,465,538,616,700,790,883,961,985,990,995,1000,1005,1009,1014,977,909,724,815,898,894,890,885,880,875,870,863,856,851,844,837,830,816,723,814,895,803,712,560,407,476,490,497,504,511,517,522,527,532,537,541,544,546,626,711,801,893,968,1017,1045,1052,1034,999,942,865,773,683,599,521,449,383,323,268,188,107,140,154,160,165,169,172,174,218,267,321,381,447,519,597,681,770,862,910,823,729,640,557,480,409,344,285,232,185,144,109,80,57,38,25,9,13,18,28,42,61,84,113,148,189,236,289,348,413,484,561,644,733,827,849,755,666,583,506,435,370,307,247,193,137,75,100,122,116,101,74,99,132,170,213,261,315,375,441,512,589,672,761,855,934,991,1026,998,936,852,751,655,565,473,339,400,458,453,448,443,437,430,423,416,401,338,399,468,542,621,705,795,888,917,923,928,933,938,943,950,955,960,965,819,907,972,971,969,967,963,958,953,947,940,935,930,925,920,914,908,818,891,798,708,623,483,342,405,419,426,433,440,445,450,455,460,464,467,469,543,622,706,796,889,966,1016,1044,1029,994,937,858,766,677,594,516,444,378,318,264,215,147,78,105,119,124,128,131,133,171,214,262,316,376,442,514,591,674,763,821,726,637,554,477,406,341,282,229,182,141,106,77,54,35,22,14,8)

Polycyclic presentation:

2,3,5,1051|c^a=c^4,d^a=d^1050,d^b=d^870

Conjugacy classes & character table

887 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31530, 17);