#36384332

order	22722
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₅₄₁ ⋊ C₂×C₃
Hall	#26015499 ⋊ C₂
Hall	D₃₇₈₇ ⋊ C₃
Sylow	C₅₄₁ ⋊ AGL(1,𝔽₇)
Sylow	C₇ ⋊ D₅₄₁⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₅₄₁	C₂×C₃
LCS	G ▷ C₇×C₅₄₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 548
transitive degree	μ_t(G) = 3787

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

Polycyclic presentation:

2,3,7,541|c^a=c^6,d^a=d^540,c^b=c^2,d^b=d^411

Conjugacy classes & character table

637 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22722, 24);