C₂×D₁₄₂₉₁ #72292208

order	57164
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₈₅₈₂
direct product	C₂×D₁₄₂₉₁

Splittings

Fitting, socle	C₂×C₃₁×C₄₆₁ ⋊ C₂
derived, Hall	C₃₁×C₄₆₁ ⋊ C₂²
Sylow	C₄₆₁ ⋊ C₂×D₃₁
Sylow	C₃₁ ⋊ C₂×D₄₆₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₂₉₁
commutator, cosocle, hypocenter	C₃₁×C₄₆₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃₁×C₄₆₁	C₂
LCS	G ▷ C₃₁×C₄₆₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 494
transitive degree	μ_t(G) = 28582

A minimal faithful permutation representation — 2 generators on 494 points:

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

Conjugacy classes & character table

14294 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×D14291 #72292208