#41008887

order	27006
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₆₄₃ ⋊ C₂×C₃
Hall	#27212595 ⋊ C₂
Hall	D₄₅₀₁ ⋊ C₃
Sylow	C₆₄₃ ⋊ AGL(1,𝔽₇)
Sylow	C₇ ⋊ D₆₄₃⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₆₄₃	C₂×C₃
LCS	G ▷ C₇×C₆₄₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 650
transitive degree	μ_t(G) = 4501

A minimal faithful permutation representation — 2 generators on 650 points:

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

Polycyclic presentation:

2,3,7,643|c^a=c^6,d^a=d^642,c^b=c^2,d^b=d^177

Conjugacy classes & character table

756 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27006, 24);