Finite Group DatabasebetaFinite Group Databasebeta| order | 31794 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C757⋊C3)⋊D7 |
| Fitting, socle, derived, Hall | C7×C757 ⋊ C2×C3 |
| Hall | C7×C757⋊C3 ⋊ C2 |
| Hall | D5299 ⋊ C3 |
| Hall | C757⋊C3 ⋊ D7 |
| Sylow | C757 ⋊ C3×D7 |
| Sylow | C7 ⋊ D757⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C757 | C2×C3 |
| LCS | G ▷ C7×C757 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 764 |
| transitive degree | μt(G) = 5299 |
A minimal faithful permutation representation — 2 generators on 764 points:
(2,3)(4,5)(6,7)(9,13,14,10,11,15)(12,21,22,16,17,31)(18,34,35,24,25,48)(19,33,29,27,23,20)(26,53,54,37,38,71)(28,51,52,40,36,45)(30,32,50,42,43,46)(39,76,77,56,57,100)(41,74,75,59,55,68)(44,49,73,61,62,69)(47,89,87,65,66,64)(58,105,106,79,80,135)(60,103,104,82,78,97)(63,72,102,84,85,98)(67,118,119,90,91,93)(70,124,116,94,95,88)(81,140,141,108,109,176)(83,138,139,111,107,132)(86,101,137,113,114,133)(92,153,154,120,121,128)(96,160,158,125,126,123)(99,165,151,129,130,117)(110,181,182,143,144,223)(112,179,180,146,142,173)(115,136,178,148,149,174)(122,194,195,155,156,169)(127,201,202,161,162,164)(131,207,199,166,167,159)(134,212,192,170,171,152)(145,228,229,184,185,276)(147,226,227,187,183,220)(150,177,225,189,190,221)(157,241,242,196,197,216)(163,248,249,203,204,211)(168,255,253,208,209,206)(172,260,246,213,214,200)(175,265,239,217,218,193)(186,281,282,231,232,335)(188,279,280,234,230,273)(191,224,278,236,237,274)(198,294,295,243,244,269)(205,301,302,250,251,264)(210,308,309,256,257,259)(215,314,306,261,262,254)(219,319,299,266,267,247)(222,324,292,270,271,240)(233,340,341,284,285,400)(235,338,339,287,283,332)(238,277,337,289,290,333)(245,353,354,296,297,328)(252,360,361,303,304,323)(258,367,368,310,311,318)(263,374,372,315,316,313)(268,379,365,320,321,307)(272,384,358,325,326,300)(275,389,351,329,330,293)(286,405,406,343,344,471)(288,403,404,346,342,397)(291,336,402,348,349,398)(298,418,419,355,356,393)(305,425,426,362,363,388)(312,432,433,369,370,383)(317,439,440,375,376,378)(322,445,437,380,381,373)(327,450,430,385,386,366)(331,455,423,390,391,359)(334,460,416,394,395,352)(345,476,477,408,409,546)(347,474,475,411,407,468)(350,401,473,413,414,469)(357,487,488,420,421,464)(364,494,495,427,428,459)(371,501,502,434,435,454)(377,508,509,441,442,449)(382,515,513,446,447,444)(387,520,506,451,452,438)(392,525,499,456,457,431)(396,530,492,461,462,424)(399,535,485,465,466,417)(410,551,552,479,480,620)(412,549,550,481,478,543)(415,472,548,482,483,544)(422,558,559,489,490,539)(429,563,564,496,497,534)(436,570,571,503,504,529)(443,577,578,510,511,524)(448,584,585,516,517,519)(453,590,582,521,522,514)(458,595,575,526,527,507)(463,600,568,531,532,500)(467,605,561,536,537,493)(470,610,556,540,541,486)(484,547,621,553,554,618)(491,622,625,555,560,614)(498,628,629,565,566,609)(505,632,633,572,573,604)(512,639,640,579,580,599)(518,646,647,586,587,594)(523,653,651,591,592,589)(528,658,644,596,597,583)(533,663,637,601,602,576)(538,668,630,606,607,569)(542,673,626,611,612,562)(545,678,623,615,616,557)(567,682,683,624,619,677)(574,686,687,634,635,672)(581,690,691,641,642,667)(588,694,695,648,649,662)(593,701,702,654,655,657)(598,707,699,659,660,652)(603,712,692,664,665,645)(608,717,688,669,670,638)(613,722,684,674,675,631)(617,726,636,679,680,627)(643,727,728,685,681,721)(650,731,732,696,697,716)(656,735,736,703,704,711)(661,739,737,708,709,706)(666,744,733,713,714,700)(671,749,729,718,719,693)(676,753,698,723,724,689)(705,754,755,730,725,748)(710,758,759,740,741,743)(715,761,756,745,746,738)(720,762,742,750,751,734)(747,763,764,757,752,760);(1,2,4,6,7,5,3)(8,9,12,18,26,39,58,81,110,145,186,233,286,345,410,481,411,346,287,234,187,146,111,82,59,40,27,11,15,20,30,44,63,86,115,150,191,238,291,350,415,484,555,489,420,355,296,243,196,155,120,90,65,43,17,23,31,45,64,88,117,152,193,240,293,352,417,486,557,624,565,496,427,362,303,250,203,161,125,94,62,25,36,46,48,68,93,123,159,200,247,300,359,424,493,562,627,634,572,503,434,369,310,256,208,166,129,85,38,55,66,69,71,97,128,164,206,254,307,366,431,500,569,631,685,641,579,510,441,375,315,261,213,170,114,57,78,91,95,98,100,132,169,211,259,313,373,438,507,576,638,689,696,648,586,516,446,380,320,266,217,149,80,107,121,126,130,133,135,173,216,264,318,378,444,514,583,645,693,730,703,654,591,521,451,385,325,270,190,109,142,156,162,167,171,174,176,220,269,323,383,449,519,589,652,700,734,740,708,659,596,526,456,390,329,237,144,183,197,204,209,214,218,221,223,273,328,388,454,524,594,657,706,738,757,745,713,664,601,531,461,394,290,185,230,244,251,257,262,267,271,274,276,332,393,459,529,599,662,711,743,760,763,750,718,669,606,536,465,349,232,283,297,304,311,316,321,326,330,333,335,397,464,534,604,667,716,748,762,761,758,754,723,674,611,540,414,285,342,356,363,370,376,381,386,391,395,398,400,468,539,609,672,721,753,749,744,739,735,731,727,679,615,483,344,407,421,428,435,442,447,452,457,462,466,469,471,543,614,677,726,722,717,712,707,701,694,690,686,682,554,409,478,490,497,504,511,517,522,527,532,537,541,544,546,552,621,678,673,668,663,658,653,646,639,632,628,622,480,551,560,566,573,580,587,592,597,602,607,612,616,618,620,477,548,610,605,600,595,590,584,577,570,563,558,549,476,547,619,635,642,649,655,660,665,670,675,680,683,625,550,406,473,535,530,525,520,515,508,501,494,487,474,405,472,545,617,681,697,704,709,714,719,724,728,687,629,559,475,341,402,460,455,450,445,439,432,425,418,403,340,401,470,542,613,676,725,741,746,751,755,732,691,633,564,488,404,282,337,389,384,379,374,367,360,353,338,281,336,399,467,538,608,671,720,752,764,759,736,695,640,571,495,419,339,229,278,324,319,314,308,301,294,279,228,277,334,396,463,533,603,666,715,747,756,737,702,647,578,502,426,354,280,182,225,265,260,255,248,241,226,181,224,275,331,392,458,528,598,661,710,742,733,699,651,585,509,433,361,295,227,141,178,212,207,201,194,179,140,177,222,272,327,387,453,523,593,656,705,729,692,644,582,513,440,368,302,242,180,106,137,165,160,153,138,105,136,175,219,268,322,382,448,518,588,650,698,688,637,575,506,437,372,309,249,195,139,77,102,124,118,103,76,101,134,172,215,263,317,377,443,512,581,643,684,630,568,499,430,365,306,253,202,154,104,54,73,89,74,53,72,99,131,168,210,258,312,371,436,505,574,636,626,561,492,423,358,299,246,199,158,119,75,35,50,51,34,49,70,96,127,163,205,252,305,364,429,498,567,623,556,485,416,351,292,239,192,151,116,87,52,22,33,21,32,47,67,92,122,157,198,245,298,357,422,491,553,482,413,348,289,236,189,148,113,84,61,42,29,14,13,19,28,41,60,83,112,147,188,235,288,347,412,479,408,343,284,231,184,143,108,79,56,37,24,16,10)Polycyclic presentation:
2,3,7,757|c^a=c^6,d^a=d^756,d^b=d^27897 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31794, 26);