Finite Group DatabasebetaFinite Group Databasebeta| order | 31830 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(D1061⋊C5) |
| unambiguous semidirect product | (C1061⋊C5)⋊D3 |
| unambiguous semidirect product | D3183⋊C5 |
| Fitting, socle, derived, Hall | C3×C1061 ⋊ C2×C5 |
| Hall | C3×C1061⋊C5 ⋊ C2 |
| Hall | D3183 ⋊ C5 |
| Hall | C1061⋊C5 ⋊ D3 |
| Sylow | C1061 ⋊ C5×D3 |
| Sylow | C3 ⋊ D1061⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C1061 | C2×C5 |
| LCS | G ▷ C3×C1061 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 1064 |
| transitive degree | μt(G) = 3183 |
A minimal faithful permutation representation — 2 generators on 1064 points:
(2,3)(5,9,10,28,33,6,7,15,17,57)(8,22,23,90,95,12,13,44,46,175)(11,38,39,40,68,19,20,78,80,82)(14,51,52,209,214,25,26,106,108,404)(16,62,63,254,235,30,31,129,131,119)(18,72,73,69,70,35,36,74,34,81)(21,84,85,265,186,41,42,162,164,358)(24,100,101,377,142,48,49,196,198,282)(27,113,114,435,437,54,55,225,227,700)(29,123,124,58,237,59,60,242,120,121)(32,136,137,488,493,65,66,269,271,338)(37,148,149,187,188,75,76,291,96,360)(43,169,170,554,413,87,88,325,327,665)(45,180,181,443,441,92,93,347,349,236)(47,190,191,143,144,97,98,365,71,284)(50,203,204,397,359,103,104,391,393,605)(53,219,220,679,283,110,111,422,424,510)(56,230,211,212,660,116,117,107,409,410)(61,248,249,176,580,126,127,461,339,340)(64,263,264,83,311,133,134,361,159,160)(67,276,277,279,280,139,140,503,141,506)(77,297,298,414,415,151,152,521,215,667)(79,305,102,386,387,155,156,200,201,238)(86,321,245,246,154,166,167,125,382,303)(89,332,333,835,702,172,173,567,569,960)(91,342,343,122,451,177,178,584,239,240)(94,354,355,528,529,183,184,600,302,579)(99,371,372,362,363,193,194,621,189,609)(105,398,399,901,666,206,207,642,644,923)(109,417,115,285,185,216,217,228,145,357)(112,429,430,947,606,222,223,688,690,596)(118,444,445,556,964,232,233,326,709,822)(128,467,468,405,278,251,252,736,655,504)(130,293,368,369,309,256,257,192,259,158)(132,367,288,289,319,260,261,147,243,165)(135,482,483,481,850,266,267,754,751,752)(138,497,498,628,536,273,274,771,772,656)(146,294,295,383,384,286,150,255,199,244)(153,526,318,703,253,300,301,545,438,470)(157,534,221,686,527,307,308,426,427,581)(161,537,351,352,668,312,313,182,558,597)(163,543,499,707,708,316,317,774,750,450)(168,550,532,205,608,322,323,306,395,396)(171,562,563,587,304,329,330,344,682,531)(174,573,574,795,921,335,336,522,843,1000)(179,589,590,341,525,345,346,857,582,583)(195,624,625,669,670,374,375,881,416,926)(197,632,633,636,716,379,380,858,616,452)(202,617,519,408,610,388,389,296,480,394)(208,649,650,1041,961,401,402,912,914,1059)(210,657,658,241,718,406,407,428,453,454)(213,593,594,722,476,412,348,863,845,746)(218,674,334,611,612,419,420,571,364,659)(224,694,695,1055,924,432,433,953,954,986)(226,490,491,418,442,310,270,523,672,231)(229,512,513,559,560,439,290,462,328,350)(234,712,713,903,981,447,448,643,965,764)(247,725,541,484,851,458,459,315,757,758)(250,732,629,630,549,464,465,378,888,811)(258,744,745,766,767,473,474,864,494,607)(262,747,689,788,852,477,478,514,585,756)(268,760,761,818,505,485,486,977,979,782)(272,720,721,436,723,495,455,613,456,701)(275,776,777,641,1020,500,501,987,898,899)(281,784,475,839,840,507,508,737,570,661)(287,524,381,683,684,511,299,489,425,586)(292,789,790,711,724,515,516,592,773,457)(314,805,662,663,471,538,539,411,838,741)(320,518,623,618,619,546,547,373,370,385)(324,814,808,778,925,551,552,544,904,985)(331,830,831,891,876,564,565,634,949,1028)(337,846,847,356,873,576,577,804,602,603)(353,868,869,753,798,598,599,952,706,974)(366,719,423,743,730,614,615,472,681,463)(376,849,886,962,487,578,627,1016,704,763)(390,715,893,821,927,449,637,791,825,988)(392,896,693,748,749,639,640,951,479,717)(400,908,699,976,533,646,647,759,941,801)(403,918,892,1029,1050,652,653,882,1030,1037)(421,934,651,928,929,676,677,916,671,1009)(431,950,887,922,535,691,692,889,813,802)(434,958,809,1063,1060,697,698,1004,991,1054)(440,794,664,905,906,705,520,755,645,775)(446,568,810,1017,673,710,548,837,883,930)(460,966,819,820,687,726,727,555,1008,945)(466,969,894,648,1021,733,734,638,909,910)(469,854,517,939,812,738,739,680,792,870)(492,968,871,540,800,729,765,601,530,806)(496,842,967,836,1022,769,728,879,853,907)(502,989,990,1033,783,779,780,1062,995,994)(509,996,867,1043,1044,785,786,978,915,1010)(542,654,980,588,998,807,762,920,855,856)(553,1005,1006,1011,742,815,816,823,1042,972)(557,1001,675,844,635,824,797,932,575,799)(561,1012,572,878,796,826,827,841,620,940)(566,1013,1014,957,984,832,833,897,1056,770)(591,731,622,890,829,859,860,631,880,862)(595,1024,1025,848,983,865,866,933,768,877)(604,1027,938,900,997,874,875,982,787,1032)(626,1023,685,975,793,884,885,942,943,999)(678,1040,959,1061,781,936,937,1058,963,992)(696,1057,1038,1039,803,955,956,911,1051,1003)(714,913,895,1046,931,861,828,1031,917,1049)(735,1015,1034,1035,946,834,970,902,993,973)(740,1007,1018,1053,1026,817,971,948,1045,872)(919,1019,1052,1064,935,1048,944,1036,1002,1047);(1,2,3)(4,5,8,14,27,56,118,234,449,388,200,100,197,381,137,272,496,770,985,597,667,784,512,288,73,72,146,287,427,271,494,768,846,772,198,382,550,813,840,660,921,1050,647,483,756,975,1007,967,372,623,371,622,629,101,199,385,635,301,527,580,850,1020,979,327,558,825,819,170,328,561,828,402,651,917,1047,934,801,532,305,284,412,211,26,53,112,224,434,699,757,482,755,662,409,184,356,604,780,991,1035,992,779,500,273,136,270,492,513,127,250,466,735,832,564,329,169,326,557,723,317,544,809,875,1011,1041,1058,958,856,939,839,567,836,788,178,344,588,637,478,615,718,805,983,363,610,391,638,895,866,1017,881,833,734,891,947,977,928,669,414,209,405,656,160,121,238,450,233,446,711,771,950,999,547,257,291,514,223,431,693,952,1054,1059,960,700,503,307,155,20,40,83,161,314,540,528,300,151,75,35,15,29,61,128,253,471,742,973,945,658,426,219,423,295,63,132,262,479,750,349,442,230,441,235,33,68,142,283,217,418,673,931,980,590,858,732,830,885,745,863,1012,1028,396,264,360,607,806,721,191,367,617,582,339,46,96,189,364,468,424,682,941,1051,1044,964,873,807,541,85,165,320,548,173,334,572,842,726,458,245,123,243,456,114,140,278,505,783,995,644,904,776,598,351,180,348,595,867,486,762,652,401,206,103,48,22,45,94,185,308,535,803,937,1038,909,646,398,643,843,968,556,823,695,599,820,1009,1032,974,583,524,297,522,796,743,380,634,892,375,626,880,966,1005,1052,1022,905,642,902,965,663,835,1016,552,459,630,889,849,1019,942,683,422,680,515,76,150,296,520,786,670,927,987,1057,1064,1001,491,521,639,104,205,397,641,900,785,507,279,139,65,30,7,11,21,43,89,174,337,578,374,193,97,44,91,179,274,499,775,986,923,665,404,506,285,143,69,28,58,122,241,455,722,844,573,724,384,242,321,549,812,355,601,872,910,758,343,586,854,878,618,368,190,366,616,201,131,259,476,529,493,95,186,359,606,560,347,592,862,330,249,463,731,653,919,1025,932,674,531,303,39,81,158,310,55,115,229,440,706,751,164,263,480,406,52,109,218,421,678,938,981,763,485,266,133,62,130,258,475,252,469,740,815,551,322,166,84,163,318,227,438,704,963,990,954,1056,955,694,893,789,744,837,1018,430,346,591,861,712,749,716,451,537,804,612,657,640,897,650,915,1045,1029,1015,970,1030,1053,1043,912,1014,896,407,659,847,313,240,452,717,448,714,859,589,688,948,810,474,516,791,433,696,957,1055,1062,1061,962,703,435,545,316,42,86,168,324,553,817,738,467,737,473,256,31,64,135,268,487,764,982,936,676,419,216,106,210,408,134,265,481,753,705,439,228,113,226,309,70,78,154,304,420,675,933,1048,918,860,730,461,562,829,790,181,350,596,605,358,175,338,579,746,369,254,386,636,614,98,192,370,620,739,684,584,851,1021,1026,871,600,870,811,167,124,244,457,336,575,845,720,453,239,120,17,34,71,145,280,214,413,666,924,906,774,497,345,177,23,47,99,195,376,576,335,172,87,41,19,9,16,32,67,141,281,509,787,898,393,395,203,392,298,523,797,1002,956,777,988,926,996,794,519,294,148,292,517,220,425,685,944,627,887,888,725,814,886,569,838,903,1034,399,645,907,1036,816,727,890,884,624,882,831,632,681,940,574,152,299,525,798,997,929,1008,868,953,1006,709,765,795,713,207,400,648,911,1040,1003,802,534,357,183,92,13,24,50,105,208,403,654,760,978,865,593,93,182,353,501,778,901,1033,994,782,504,276,225,436,289,60,125,247,460,728,841,571,332,568,546,319,162,315,542,603,822,1010,1039,976,587,679,736,611,362,187,90,176,341,389,261,365,613,800,767,188,361,608,876,827,594,864,1023,565,465,633,857,920,1049,930,672,417,510,282,82,57,119,236,117,231,443,707,748,477,260,129,255,472,111,221,428,687,946,972,741,470,251,126,59,10,18,37,77,153,302,530,538,312,159,80,38,79,157,277,437,702,961,1060,869,951,691,429,689,149,293,518,793,692,498,773,710,444,708,387,237,311,536,655,108,215,416,671,761,690,949,969,1013,625,883,1024,1031,894,204,394,609,877,539,454,719,747,715,855,563,342,585,853,333,570,792,998,698,959,914,1042,1027,1004,808,543,701,824,445,88,171,331,566,834,733,464,248,462,729,490,66,138,275,502,781,993,1063,989,874,602,354,212,411,664,267,484,759,697,432,222,110,51,107,213,144,156,306,533,677,935,1046,916,649,913,826,559,325,555,821,352,554,818,899,752,340,581,526,799,619,383,196,378,631,194,373,621,879,971,943,852,754,908,1037,1000,410,661,922,323,246,377,628,577,848,766,488,686,511,286,36,74,147,290,508,415,668,925,984,769,495,269,489,379,49,102,202,390,447,232,116,54,25,12,6)Polycyclic presentation:
2,3,5,1061|b^a=b^2,d^a=d^1060,d^c=d^381333 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31830, 12);