#45682976

order	33054
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₇₈₇ ⋊ C₂×C₃
Hall	#30658283 ⋊ C₂
Hall	D₅₅₀₉ ⋊ C₃
Sylow	C₇₈₇ ⋊ AGL(1,𝔽₇)
Sylow	C₇ ⋊ D₇₈₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₇₈₇	C₂×C₃
LCS	G ▷ C₇×C₇₈₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 794
transitive degree	μ_t(G) = 5509

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

Polycyclic presentation:

2,3,7,787|c^a=c^6,d^a=d^786,c^b=c^2,d^b=d^379

Conjugacy classes & character table

924 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33054, 23);