Finite Group DatabasebetaFinite Group Databasebeta| order | 29778 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C709⋊C3)⋊D7 |
| Fitting, socle, derived, Hall | C7×C709 ⋊ C2×C3 |
| Hall | C7×C709⋊C3 ⋊ C2 |
| Hall | D4963 ⋊ C3 |
| Hall | C709⋊C3 ⋊ D7 |
| Sylow | C709 ⋊ C3×D7 |
| Sylow | C7 ⋊ D709⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C709 | C2×C3 |
| LCS | G ▷ C7×C709 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 716 |
| transitive degree | μt(G) = 4963 |
A minimal faithful permutation representation — 2 generators on 716 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)(345,476,477,408,409,547)(347,474,475,411,407,468)(350,401,473,413,414,469)(357,489,490,420,421,464)(364,496,497,427,428,459)(371,503,504,434,435,454)(377,510,511,441,442,449)(382,517,515,446,447,444)(387,522,508,451,452,438)(392,527,501,456,457,431)(396,532,494,461,462,424)(399,537,487,465,466,417)(410,552,553,479,480,616)(412,550,551,482,478,544)(415,472,549,484,485,545)(422,559,560,491,492,541)(429,564,565,498,499,536)(436,571,572,505,506,531)(443,578,579,512,513,526)(448,585,586,518,519,521)(453,591,583,523,524,516)(458,596,576,528,529,509)(463,601,569,533,534,502)(467,606,481,538,539,495)(470,611,558,542,543,488)(483,617,618,555,554,614)(486,548,613,557,546,615)(493,619,620,561,562,612)(500,623,624,566,567,610)(507,628,629,573,574,605)(514,633,634,580,581,600)(520,640,641,587,588,595)(525,647,645,592,593,590)(530,652,638,597,598,584)(535,657,622,602,603,577)(540,662,556,607,608,570)(563,663,664,621,609,667)(568,668,669,625,626,666)(575,670,671,630,631,665)(582,674,675,635,636,661)(589,679,680,642,643,656)(594,684,685,648,649,651)(599,690,673,653,654,646)(604,695,627,658,659,639)(632,696,697,672,660,700)(637,701,702,676,677,699)(644,703,704,681,682,698)(650,707,706,686,687,694)(655,711,678,691,692,689)(683,712,713,705,693,715)(688,716,710,708,709,714);(1,2,4,6,7,5,3)(8,9,12,18,26,39,58,81,110,145,186,233,286,345,410,481,494,423,358,299,246,199,158,119,75,35,50,51,34,49,70,96,127,163,205,252,305,364,429,500,568,627,638,583,515,440,368,302,242,180,106,137,165,160,153,138,105,136,175,219,268,322,382,448,520,589,644,683,706,680,634,572,497,419,339,229,278,324,319,314,308,301,294,279,228,277,334,396,463,535,604,660,677,682,687,692,697,669,620,551,406,473,537,532,527,522,517,510,503,496,489,474,405,472,546,409,478,492,499,506,513,519,524,529,534,539,543,545,547,614,667,695,690,684,679,674,670,668,663,607,538,465,349,232,283,297,304,311,316,321,326,330,333,335,397,464,536,605,661,698,714,708,686,648,592,523,451,385,325,270,190,109,142,156,162,167,171,174,176,220,269,323,383,449,521,590,646,672,630,573,505,434,369,310,256,208,166,129,85,38,55,66,69,71,97,128,164,206,254,307,366,431,502,570,555,482,411,346,287,234,187,146,111,82,59,40,27,11,15,20,30,44,63,86,115,150,191,238,291,350,415,486,558,487,416,351,292,239,192,151,116,87,52,22,33,21,32,47,67,92,122,157,198,245,298,357,422,493,563,622,576,508,437,372,309,249,195,139,77,102,124,118,103,76,101,134,172,215,263,317,377,443,514,582,637,678,685,641,579,504,426,354,280,182,225,265,260,255,248,241,226,181,224,275,331,392,458,530,599,655,693,709,713,702,671,624,560,475,341,402,460,455,450,445,439,432,425,418,403,340,401,470,480,554,562,567,574,581,588,593,598,603,608,553,613,615,616,662,657,652,647,640,633,628,623,619,617,552,542,414,285,342,356,363,370,376,381,386,391,395,398,400,468,541,610,665,699,715,716,712,691,653,597,528,456,390,329,237,144,183,197,204,209,214,218,221,223,273,328,388,454,526,595,651,689,676,635,580,512,441,375,315,261,213,170,114,57,78,91,95,98,100,132,169,211,259,313,373,438,509,577,621,561,491,420,355,296,243,196,155,120,90,65,43,17,23,31,45,64,88,117,152,193,240,293,352,417,488,557,484,413,348,289,236,189,148,113,84,61,42,29,14,13,19,28,41,60,83,112,147,188,235,288,347,412,483,556,569,501,430,365,306,253,202,154,104,54,73,89,74,53,72,99,131,168,210,258,312,371,436,507,575,632,673,645,586,511,433,361,295,227,141,178,212,207,201,194,179,140,177,222,272,327,387,453,525,594,650,688,710,704,675,629,565,490,404,282,337,389,384,379,374,367,360,353,338,281,336,399,467,540,609,626,631,636,643,649,654,659,664,618,477,549,611,606,601,596,591,585,578,571,564,559,550,476,548,485,344,407,421,428,435,442,447,452,457,462,466,469,471,544,612,666,700,711,707,703,701,696,658,602,533,461,394,290,185,230,244,251,257,262,267,271,274,276,332,393,459,531,600,656,694,705,681,642,587,518,446,380,320,266,217,149,80,107,121,126,130,133,135,173,216,264,318,378,444,516,584,639,625,566,498,427,362,303,250,203,161,125,94,62,25,36,46,48,68,93,123,159,200,247,300,359,424,495,479,408,343,284,231,184,143,108,79,56,37,24,16,10)Polycyclic presentation:
2,3,7,709|c^a=c^6,d^a=d^708,d^b=d^227841 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(29778, 21);