#45682977

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	#30658284 ⋊ 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)(422,563,564,491,492,541)(429,568,569,498,499,536)(436,575,576,505,506,531)(443,582,583,512,513,526)(448,589,590,518,519,521)(453,595,587,523,524,516)(458,600,580,528,529,509)(463,605,573,533,534,502)(467,610,566,538,539,495)(470,615,561,542,543,488)(481,565,619,493,556,634)(483,629,630,557,555,623)(486,549,628,558,559,624)(500,637,638,570,571,614)(507,641,642,577,578,609)(514,648,649,584,585,604)(520,655,656,591,592,599)(525,662,660,596,597,594)(530,667,653,601,602,588)(535,672,646,606,607,581)(540,677,639,611,612,574)(544,682,635,616,617,567)(547,687,632,620,621,562)(560,627,686,572,631,694)(579,699,700,643,644,681)(586,703,704,650,651,676)(593,707,708,657,658,671)(598,714,715,663,664,666)(603,720,712,668,669,661)(608,725,705,673,674,654)(613,730,701,678,679,647)(618,735,697,683,684,640)(622,740,695,688,689,636)(625,739,645,691,692,633)(652,744,745,696,693,734)(659,750,751,709,710,729)(665,754,755,716,717,724)(670,758,756,721,722,719)(675,763,752,726,727,713)(680,768,748,731,732,706)(685,773,746,736,737,702)(690,772,711,741,742,698)(718,777,778,747,743,767)(723,783,784,759,760,762)(728,786,781,764,765,757)(733,788,779,769,770,753)(738,787,761,774,775,749)(766,792,793,780,776,785)(771,794,791,789,790,782);(1,2,4,7,6,5,3)(8,9,12,18,26,39,58,81,110,145,186,233,286,345,410,481,491,420,355,296,243,196,155,120,90,65,43,17,23,31,45,64,88,117,152,193,240,293,352,417,488,562,633,643,577,505,434,369,310,256,208,166,129,85,38,55,66,69,71,97,128,164,206,254,307,366,431,502,574,640,698,709,657,591,518,446,380,320,266,217,149,80,107,121,126,130,133,135,173,216,264,318,378,444,516,588,654,706,749,759,721,668,601,528,456,390,329,237,144,183,197,204,209,214,218,221,223,273,328,388,454,526,599,666,719,757,782,789,769,731,678,611,538,465,349,232,283,297,304,311,316,321,326,330,333,335,397,464,536,609,676,729,767,787,788,786,783,777,741,688,620,485,344,407,421,428,435,442,447,452,457,462,466,469,471,545,619,686,739,740,735,730,725,720,714,707,703,699,631,480,555,565,571,578,585,592,597,602,607,612,617,621,624,626,630,477,550,615,610,605,600,595,589,582,575,568,563,551,476,549,625,693,710,717,722,727,732,737,742,745,700,638,564,475,341,402,460,455,450,445,439,432,425,418,403,340,401,470,544,618,685,738,776,790,793,784,755,708,649,576,497,419,339,229,278,324,319,314,308,301,294,279,228,277,334,396,463,535,608,675,728,766,779,752,712,660,590,511,433,361,295,227,141,178,212,207,201,194,179,140,177,222,272,327,387,453,525,598,665,718,746,701,646,580,508,437,372,309,249,195,139,77,102,124,118,103,76,101,134,172,215,263,317,377,443,514,586,652,695,635,566,494,423,358,299,246,199,158,119,75,35,50,51,34,49,70,96,127,163,205,252,305,364,429,500,572,558,484,413,348,289,236,189,148,113,84,61,42,29,14,13,19,28,41,60,83,112,147,188,235,288,347,412,483,557,482,411,346,287,234,187,146,111,82,59,40,27,11,15,20,30,44,63,86,115,150,191,238,291,350,415,486,560,570,498,427,362,303,250,203,161,125,94,62,25,36,46,48,68,93,123,159,200,247,300,359,424,495,567,636,696,650,584,512,441,375,315,261,213,170,114,57,78,91,95,98,100,132,169,211,259,313,373,438,509,581,647,702,747,716,663,596,523,451,385,325,270,190,109,142,156,162,167,171,174,176,220,269,323,383,449,521,594,661,713,753,780,764,726,673,606,533,461,394,290,185,230,244,251,257,262,267,271,274,276,332,393,459,531,604,671,724,762,785,794,792,774,736,683,616,542,414,285,342,356,363,370,376,381,386,391,395,398,400,468,541,614,681,734,772,773,768,763,758,754,750,744,691,559,409,478,492,499,506,513,519,524,529,534,539,543,546,548,623,554,628,687,682,677,672,667,662,655,648,641,637,556,629,553,627,644,651,658,664,669,674,679,684,689,692,694,634,552,406,473,537,532,527,522,517,510,503,496,489,474,405,472,547,622,690,743,760,765,770,775,778,751,704,642,569,490,404,282,337,389,384,379,374,367,360,353,338,281,336,399,467,540,613,680,733,771,791,781,756,715,656,583,504,426,354,280,182,225,265,260,255,248,241,226,181,224,275,331,392,458,530,603,670,723,761,748,705,653,587,515,440,368,302,242,180,106,137,165,160,153,138,105,136,175,219,268,322,382,448,520,593,659,711,697,639,573,501,430,365,306,253,202,154,104,54,73,89,74,53,72,99,131,168,210,258,312,371,436,507,579,645,632,561,487,416,351,292,239,192,151,116,87,52,22,33,21,32,47,67,92,122,157,198,245,298,357,422,493,479,408,343,284,231,184,143,108,79,56,37,24,16,10)

Polycyclic presentation:

2,3,7,787|c^a=c^6,d^a=d^786,c^b=c^2,d^b=d^407

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, 24);