Finite Group DatabasebetaFinite Group Databasebeta| order | 29791 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C312 · Schur multiplier (H2) = C312 |
| unitriangular | UT(3,𝔽31) |
| extraspecial | 31+1+2 |
| Heisenberg | H(3,𝔽31) |
| unambiguous semidirect product | C312⋊C31 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C31 | C312 |
| LCS | G ▷ C31 ▷ C1 | |
| UCS | C1 ◁ C31 ◁ G | |
| LECS | G ▷ C31 ▷ C1 | |
67 conjugacy classes of subgroups.
| minimal degree | μ(G) = 961 |
| transitive degree | μt(G) = 961 |
A minimal faithful permutation representation — 2 generators on 961 points:
(1,2,4,8,15,26,42,64,93,130,176,232,279,322,359,396,433,470,503,536,569,602,635,668,701,734,767,800,833,866,899)(3,6,10,17,28,44,66,95,132,178,234,281,324,361,398,435,472,505,538,571,604,637,670,703,736,769,802,835,868,901,932)(5,11,21,37,59,88,125,171,227,274,317,355,392,429,466,501,534,567,600,633,666,699,732,765,798,831,864,897,930,961,931)(7,13,20,31,47,69,98,135,181,237,284,327,364,401,438,474,507,540,573,606,639,672,705,738,771,804,837,870,903,900,933)(9,18,32,53,83,120,166,222,270,314,354,391,428,465,500,533,566,599,632,665,698,731,764,797,830,863,896,929,960,898,12)(14,24,36,52,74,103,140,186,242,289,331,368,405,442,437,476,510,544,578,612,646,680,714,748,782,816,850,884,867,902,934)(16,29,48,75,112,160,216,268,313,353,390,427,464,499,532,565,598,631,664,697,730,763,796,829,862,895,928,959,865,23,22)(19,34,27,45,70,104,149,207,267,312,352,389,426,463,498,531,564,597,630,663,696,729,762,795,828,861,894,927,958,832,39)(25,40,58,82,111,148,194,249,296,288,335,375,415,455,440,478,512,546,580,614,648,682,716,750,784,818,852,834,869,904,935)(30,50,78,65,96,136,187,250,303,348,386,424,461,496,529,562,595,628,661,694,727,760,793,826,859,892,925,956,766,90,35)(33,54,43,67,99,141,195,260,310,351,388,425,462,497,530,563,596,629,662,695,728,761,794,827,860,893,926,957,799,61,38)(41,62,87,119,159,206,259,255,308,292,338,378,418,400,441,479,513,547,581,615,649,683,717,751,785,819,801,836,871,905,936)(46,72,107,153,131,179,238,290,336,376,416,456,492,526,560,593,626,659,692,725,758,791,824,857,890,923,954,700,173,57,55)(49,76,113,94,133,182,243,297,342,382,421,459,495,528,561,594,627,660,693,726,759,792,825,858,891,924,955,733,127,60,56)(51,80,71,105,150,208,177,235,285,332,372,412,452,489,523,557,591,625,658,691,724,757,790,823,856,889,922,953,667,229,89)(63,91,124,165,215,205,266,257,309,294,340,380,420,403,444,481,515,549,583,617,651,685,719,753,787,768,803,838,872,906,937)(68,101,144,199,264,233,282,328,369,409,449,486,520,554,588,622,656,690,723,756,789,822,855,888,921,952,634,276,86,85,79)(73,109,97,138,190,253,306,350,323,362,402,443,480,514,548,582,616,650,684,718,752,786,820,853,886,919,950,568,356,123,81)(77,114,100,142,196,261,311,280,325,365,406,446,483,517,551,585,619,653,687,721,755,788,821,854,887,920,951,601,319,126,84)(92,128,170,221,214,210,193,258,241,295,341,381,363,404,445,482,516,550,584,618,652,686,720,754,735,770,805,839,873,907,938)(102,146,202,183,244,298,343,383,422,460,434,473,508,542,576,610,644,678,712,746,780,814,848,882,916,947,469,467,228,118,115)(106,151,134,184,246,300,345,385,423,397,436,475,509,543,577,611,645,679,713,747,781,815,849,883,917,948,502,430,169,121,116)(108,154,137,188,251,304,349,387,360,399,439,477,511,545,579,613,647,681,715,749,783,817,851,885,918,949,535,393,172,122,117)(110,157,145,200,180,240,293,339,379,419,458,494,471,506,541,575,609,643,677,711,745,779,813,847,881,915,946,432,468,225,168)(129,174,226,269,217,211,198,263,245,299,344,384,366,407,447,484,518,552,586,620,654,688,722,702,737,772,806,840,874,908,939)(139,192,256,236,287,334,374,414,454,491,525,559,537,572,607,641,675,709,743,777,811,845,879,913,944,358,394,272,164,162,155)(143,197,262,239,291,337,377,417,457,493,527,504,539,574,608,642,676,710,744,778,812,846,880,914,945,395,431,275,167,163,156)(147,204,189,252,305,283,330,371,411,451,488,522,556,590,624,603,638,673,707,741,775,809,843,877,911,942,278,320,315,223,158)(152,209,191,254,307,286,333,373,413,453,490,524,558,592,570,605,640,674,708,742,776,810,844,878,912,943,321,357,318,224,161)(175,230,273,271,218,212,201,265,247,301,346,326,367,408,448,485,519,553,587,621,655,689,669,704,739,773,807,841,875,909,940)(185,248,302,347,329,370,410,450,487,521,555,589,623,657,636,671,706,740,774,808,842,876,910,941,231,277,316,220,219,213,203);(2,5,12,23,39,61,90,127,173,229,276,319,356,393,430,467,468,431,394,357,320,277,230,174,128,91,62,40,24,13,6)(4,9,19,35,57,86,123,169,225,272,315,273,170,87,36,10,11,22,38,60,89,126,172,228,275,318,316,226,124,58,20)(8,16,30,51,81,118,164,220,221,82,17,18,33,55,84,121,167,223,269,119,31,21,34,56,85,122,168,224,271,165,52)(15,27,46,73,110,158,214,74,37,54,80,117,163,219,215,47,32,50,79,116,162,218,159,28,29,49,77,115,161,217,111)(26,43,68,102,147,205,44,45,71,106,152,210,69,48,72,108,155,211,103,53,76,109,156,212,148,59,78,114,157,213,206)(42,65,97,139,193,66,67,100,143,198,98,70,101,145,201,140,75,105,146,203,194,83,107,151,204,259,88,113,154,209,266)(64,94,134,185,186,104,142,192,257,125,153,202,265,135,99,138,191,255,120,150,200,263,95,96,137,189,249,112,144,197,258)(93,131,180,241,171,208,262,309,166,199,256,308,160,196,254,296,149,190,252,242,141,188,248,181,136,184,247,132,133,183,245)(130,177,236,288,195,246,299,227,264,307,289,187,244,295,222,261,305,237,182,240,294,216,253,302,178,179,239,292,207,251,301)(176,233,283,234,235,286,284,238,287,331,243,291,335,250,293,338,260,298,340,267,300,341,268,304,344,270,306,346,274,311,347)(232,280,326,314,349,381,312,343,378,303,337,368,290,333,281,282,329,317,350,384,313,345,380,310,339,375,297,334,327,285,330)(279,323,363,351,377,364,328,367,353,383,415,336,371,355,387,420,348,374,324,325,366,352,379,405,332,370,354,385,418,342,373)(322,360,400,376,410,390,419,401,365,404,386,413,392,423,455,372,408,389,417,361,362,403,382,411,391,422,442,369,407,388,414)(359,397,437,406,444,416,448,425,453,428,458,398,399,440,409,445,421,450,426,454,429,460,438,402,441,412,447,424,451,427,457)(396,434,435,436,474,439,476,443,478,446,479,449,481,452,482,456,484,459,485,461,487,462,488,463,490,464,491,465,493,466,494)(433,471,501,527,500,525,499,524,498,522,497,521,496,519,495,518,492,516,489,515,486,513,483,512,480,510,477,507,475,472,473)(470,504,532,556,529,552,523,547,514,540,508,534,559,531,555,528,550,520,546,511,505,506,533,558,530,553,526,549,517,544,509)(503,537,563,586,554,578,542,566,590,561,583,548,538,539,564,587,557,580,543,567,592,562,584,551,573,541,565,589,560,581,545)(536,570,594,615,577,599,623,591,612,575,597,620,585,571,572,595,617,579,600,624,593,614,576,598,621,588,606,574,596,618,582)(569,603,625,639,607,627,648,609,629,651,611,631,654,616,633,657,622,604,605,626,646,608,628,649,610,630,652,613,632,655,619)(602,636,653,666,689,650,665,688,647,664,686,645,663,685,644,662,683,643,661,682,642,660,680,641,659,672,640,658,637,638,656)(635,669,681,696,717,676,692,670,671,684,697,719,677,693,705,673,687,698,720,678,694,714,674,690,699,722,679,695,716,675,691)(668,702,712,726,703,704,713,727,738,706,715,728,748,707,718,729,750,708,721,730,751,709,723,731,753,710,724,732,754,711,725)(701,735,744,756,763,784,741,749,760,736,737,745,757,764,785,742,752,761,771,739,746,758,765,787,743,755,762,782,740,747,759)(734,768,776,783,792,798,819,775,781,791,797,818,774,780,790,796,816,773,779,789,795,804,772,778,788,794,769,770,777,786,793)(767,801,808,813,821,826,831,852,807,812,820,825,830,850,806,811,817,824,829,837,805,810,815,823,828,802,803,809,814,822,827)(800,834,840,844,848,854,858,862,835,836,841,845,849,855,859,863,870,838,842,846,851,856,860,864,884,839,843,847,853,857,861)(833,867,872,875,878,881,885,888,891,894,897,903,871,874,877,880,883,887,890,893,896,868,869,873,876,879,882,886,889,892,895)(866,900,904,906,908,910,912,914,916,918,920,922,924,926,928,930,901,902,905,907,909,911,913,915,917,919,921,923,925,927,929)(899,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961)Polycyclic presentation:
31^3|b^a=bc991 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(29791, 3);