#44541200

order	31794
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₇₅₇ ⋊ C₂×C₃
Hall	#30629660 ⋊ C₂
Hall	D₅₂₉₉ ⋊ C₃
Sylow	C₇₅₇ ⋊ AGL(1,𝔽₇)
Sylow	C₇ ⋊ D₇₅₇⋊C₃

Subgroups

	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.

Full subgroup data →

Presentation

minimal degree	μ(G) = 764
transitive degree	μ_t(G) = 5299

A minimal faithful permutation representation — 2 generators on 764 points:

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

Polycyclic presentation:

2,3,7,757|c^a=c^6,d^a=d^756,c^b=c^2,d^b=d^729

Conjugacy classes & character table

889 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31794, 29);