#37525890

order	22974
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₇×C₅₄₇ ⋊ C₂×C₃
Hall	#26039867 ⋊ C₂
Hall	D₃₈₂₉ ⋊ C₃
Sylow	C₅₄₇ ⋊ AGL(1,𝔽₇)
Sylow	C₇ ⋊ D₅₄₇⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₅₄₇	C₂×C₃
LCS	G ▷ C₇×C₅₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 554
transitive degree	μ_t(G) = 3829

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

Polycyclic presentation:

2,3,7,547|c^a=c^6,d^a=d^546,c^b=c^2,d^b=d^40

Conjugacy classes & character table

644 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22974, 28);