#46828950

order	34566
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₈₂₃ ⋊ C₂×C₃
Hall	#31772217 ⋊ C₂
Hall	D₅₇₆₁ ⋊ C₃
Sylow	C₈₂₃ ⋊ AGL(1,𝔽₇)
Sylow	C₇ ⋊ D₈₂₃⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₈₂₃	C₂×C₃
LCS	G ▷ C₇×C₈₂₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 830
transitive degree	μ_t(G) = 5761

A minimal faithful permutation representation — 2 generators on 830 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)(345,476,477,408,409,548)(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)(410,553,554,479,480,631)(412,551,552,482,478,545)(415,472,550,484,485,546)(422,566,567,491,492,541)(429,573,574,498,499,536)(436,580,581,505,506,531)(443,587,588,512,513,526)(448,594,595,518,519,521)(453,600,592,523,524,516)(458,605,585,528,529,509)(463,610,578,533,534,502)(467,615,571,538,539,495)(470,620,564,542,543,488)(481,636,637,556,557,720)(483,634,635,559,555,628)(486,549,633,561,562,629)(493,649,650,568,569,624)(500,656,657,575,576,619)(507,663,664,582,583,614)(514,670,671,589,590,609)(520,677,678,596,597,604)(525,684,682,601,602,599)(530,689,675,606,607,593)(535,694,668,611,612,586)(540,699,661,616,617,579)(544,704,654,621,622,572)(547,709,647,625,626,565)(558,725,726,639,640,799)(560,723,724,642,638,717)(563,632,722,644,645,718)(570,736,737,651,652,713)(577,743,744,658,659,708)(584,749,750,665,666,703)(591,755,756,672,673,698)(598,762,763,679,680,693)(603,769,770,685,686,688)(608,775,767,690,691,683)(613,780,760,695,696,676)(618,785,753,700,701,669)(623,788,747,705,706,662)(627,791,741,710,711,655)(630,794,734,714,715,648)(641,738,739,728,653,806)(643,801,802,729,727,797)(646,721,800,731,732,798)(660,811,733,745,746,793)(667,814,805,751,752,790)(674,817,810,757,758,787)(681,821,813,764,765,784)(687,824,816,771,772,779)(692,826,820,776,777,774)(697,828,823,781,782,768)(702,829,819,783,786,761)(707,827,815,778,789,754)(712,825,812,773,792,748)(716,822,809,766,795,742)(719,818,804,759,796,735)(730,807,808,803,740,830);(1,2,4,7,6,5,3)(8,9,12,18,26,39,58,81,110,145,186,233,286,345,410,481,558,641,724,554,633,709,704,699,694,689,684,677,670,663,656,649,634,553,632,719,757,672,589,512,441,375,315,261,213,170,114,57,78,91,95,98,100,132,169,211,259,313,373,438,509,586,669,754,816,763,671,581,497,419,339,229,278,324,319,314,308,301,294,279,228,277,334,396,463,535,613,697,783,700,616,538,465,349,232,283,297,304,311,316,321,326,330,333,335,397,464,536,614,698,784,812,747,661,578,501,430,365,306,253,202,154,104,54,73,89,74,53,72,99,131,168,210,258,312,371,436,507,584,667,732,557,638,652,659,666,673,680,686,691,696,701,706,711,715,718,720,797,803,729,642,559,482,411,346,287,234,187,146,111,82,59,40,27,11,15,20,30,44,63,86,115,150,191,238,291,350,415,486,563,646,733,737,635,477,550,620,615,610,605,600,594,587,580,573,566,551,476,549,630,716,764,679,596,518,446,380,320,266,217,149,80,107,121,126,130,133,135,173,216,264,318,378,444,516,593,676,761,820,770,678,588,504,426,354,280,182,225,265,260,255,248,241,226,181,224,275,331,392,458,530,608,692,778,705,621,542,414,285,342,356,363,370,376,381,386,391,395,398,400,468,541,619,703,787,809,741,654,571,494,423,358,299,246,199,158,119,75,35,50,51,34,49,70,96,127,163,205,252,305,364,429,500,577,660,640,727,739,746,752,758,765,772,777,782,786,789,792,795,796,798,799,830,807,738,651,568,491,420,355,296,243,196,155,120,90,65,43,17,23,31,45,64,88,117,152,193,240,293,352,417,488,565,648,735,805,744,650,552,406,473,537,532,527,522,517,510,503,496,489,474,405,472,547,627,712,771,685,601,523,451,385,325,270,190,109,142,156,162,167,171,174,176,220,269,323,383,449,521,599,683,768,823,767,682,595,511,433,361,295,227,141,178,212,207,201,194,179,140,177,222,272,327,387,453,525,603,687,773,710,625,485,344,407,421,428,435,442,447,452,457,462,466,469,471,545,624,708,790,804,734,647,564,487,416,351,292,239,192,151,116,87,52,22,33,21,32,47,67,92,122,157,198,245,298,357,422,493,570,653,740,808,726,800,818,822,825,827,829,828,826,824,821,817,814,811,806,801,725,745,658,575,498,427,362,303,250,203,161,125,94,62,25,36,46,48,68,93,123,159,200,247,300,359,424,495,572,655,742,810,750,657,567,475,341,402,460,455,450,445,439,432,425,418,403,340,401,470,544,623,707,776,690,606,528,456,390,329,237,144,183,197,204,209,214,218,221,223,273,328,388,454,526,604,688,774,819,760,675,592,515,440,368,302,242,180,106,137,165,160,153,138,105,136,175,219,268,322,382,448,520,598,681,766,714,562,409,478,492,499,506,513,519,524,529,534,539,543,546,548,628,713,793,731,644,561,484,413,348,289,236,189,148,113,84,61,42,29,14,13,19,28,41,60,83,112,147,188,235,288,347,412,483,560,643,730,802,637,722,794,791,788,785,780,775,769,762,755,749,743,736,723,636,721,751,665,582,505,434,369,310,256,208,166,129,85,38,55,66,69,71,97,128,164,206,254,307,366,431,502,579,662,748,813,756,664,574,490,404,282,337,389,384,379,374,367,360,353,338,281,336,399,467,540,618,702,781,695,611,533,461,394,290,185,230,244,251,257,262,267,271,274,276,332,393,459,531,609,693,779,815,753,668,585,508,437,372,309,249,195,139,77,102,124,118,103,76,101,134,172,215,263,317,377,443,514,591,674,759,645,480,555,569,576,583,590,597,602,607,612,617,622,626,629,631,717,728,639,556,479,408,343,284,231,184,143,108,79,56,37,24,16,10)

Polycyclic presentation:

2,3,7,823|c^a=c^6,d^a=d^822,c^b=c^2,d^b=d^174

Conjugacy classes & character table

966 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34566, 24);