C₂×D₁₆₂₁₃ #72302129

order	64852
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₃₂₄₂₆
direct product	C₂×D₁₆₂₁₃

Splittings

Fitting, socle	C₂×C₃₁×C₅₂₃ ⋊ C₂
derived, Hall	C₃₁×C₅₂₃ ⋊ C₂²
Sylow	C₅₂₃ ⋊ C₂×D₃₁
Sylow	C₃₁ ⋊ C₂×D₅₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₆₂₁₃
commutator, cosocle, hypocenter	C₃₁×C₅₂₃	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃₁×C₅₂₃	C₂
LCS	G ▷ C₃₁×C₅₂₃
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 556
transitive degree	μ_t(G) = 32426

A minimal faithful permutation representation — 2 generators on 556 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556);(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)(35,556)(36,555)(37,554)(38,553)(39,552)(40,551)(41,550)(42,549)(43,548)(44,547)(45,546)(46,545)(47,544)(48,543)(49,542)(50,541)(51,540)(52,539)(53,538)(54,537)(55,536)(56,535)(57,534)(58,533)(59,532)(60,531)(61,530)(62,529)(63,528)(64,527)(65,526)(66,525)(67,524)(68,523)(69,522)(70,521)(71,520)(72,519)(73,518)(74,517)(75,516)(76,515)(77,514)(78,513)(79,512)(80,511)(81,510)(82,509)(83,508)(84,507)(85,506)(86,505)(87,504)(88,503)(89,502)(90,501)(91,500)(92,499)(93,498)(94,497)(95,496)(96,495)(97,494)(98,493)(99,492)(100,491)(101,490)(102,489)(103,488)(104,487)(105,486)(106,485)(107,484)(108,483)(109,482)(110,481)(111,480)(112,479)(113,478)(114,477)(115,476)(116,475)(117,474)(118,473)(119,472)(120,471)(121,470)(122,469)(123,468)(124,467)(125,466)(126,465)(127,464)(128,463)(129,462)(130,461)(131,460)(132,459)(133,458)(134,457)(135,456)(136,455)(137,454)(138,453)(139,452)(140,451)(141,450)(142,449)(143,448)(144,447)(145,446)(146,445)(147,444)(148,443)(149,442)(150,441)(151,440)(152,439)(153,438)(154,437)(155,436)(156,435)(157,434)(158,433)(159,432)(160,431)(161,430)(162,429)(163,428)(164,427)(165,426)(166,425)(167,424)(168,423)(169,422)(170,421)(171,420)(172,419)(173,418)(174,417)(175,416)(176,415)(177,414)(178,413)(179,412)(180,411)(181,410)(182,409)(183,408)(184,407)(185,406)(186,405)(187,404)(188,403)(189,402)(190,401)(191,400)(192,399)(193,398)(194,397)(195,396)(196,395)(197,394)(198,393)(199,392)(200,391)(201,390)(202,389)(203,388)(204,387)(205,386)(206,385)(207,384)(208,383)(209,382)(210,381)(211,380)(212,379)(213,378)(214,377)(215,376)(216,375)(217,374)(218,373)(219,372)(220,371)(221,370)(222,369)(223,368)(224,367)(225,366)(226,365)(227,364)(228,363)(229,362)(230,361)(231,360)(232,359)(233,358)(234,357)(235,356)(236,355)(237,354)(238,353)(239,352)(240,351)(241,350)(242,349)(243,348)(244,347)(245,346)(246,345)(247,344)(248,343)(249,342)(250,341)(251,340)(252,339)(253,338)(254,337)(255,336)(256,335)(257,334)(258,333)(259,332)(260,331)(261,330)(262,329)(263,328)(264,327)(265,326)(266,325)(267,324)(268,323)(269,322)(270,321)(271,320)(272,319)(273,318)(274,317)(275,316)(276,315)(277,314)(278,313)(279,312)(280,311)(281,310)(282,309)(283,308)(284,307)(285,306)(286,305)(287,304)(288,303)(289,302)(290,301)(291,300)(292,299)(293,298)(294,297)(295,296)

Conjugacy classes & character table

16216 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556), (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)(35,556)(36,555)(37,554)(38,553)(39,552)(40,551)(41,550)(42,549)(43,548)(44,547)(45,546)(46,545)(47,544)(48,543)(49,542)(50,541)(51,540)(52,539)(53,538)(54,537)(55,536)(56,535)(57,534)(58,533)(59,532)(60,531)(61,530)(62,529)(63,528)(64,527)(65,526)(66,525)(67,524)(68,523)(69,522)(70,521)(71,520)(72,519)(73,518)(74,517)(75,516)(76,515)(77,514)(78,513)(79,512)(80,511)(81,510)(82,509)(83,508)(84,507)(85,506)(86,505)(87,504)(88,503)(89,502)(90,501)(91,500)(92,499)(93,498)(94,497)(95,496)(96,495)(97,494)(98,493)(99,492)(100,491)(101,490)(102,489)(103,488)(104,487)(105,486)(106,485)(107,484)(108,483)(109,482)(110,481)(111,480)(112,479)(113,478)(114,477)(115,476)(116,475)(117,474)(118,473)(119,472)(120,471)(121,470)(122,469)(123,468)(124,467)(125,466)(126,465)(127,464)(128,463)(129,462)(130,461)(131,460)(132,459)(133,458)(134,457)(135,456)(136,455)(137,454)(138,453)(139,452)(140,451)(141,450)(142,449)(143,448)(144,447)(145,446)(146,445)(147,444)(148,443)(149,442)(150,441)(151,440)(152,439)(153,438)(154,437)(155,436)(156,435)(157,434)(158,433)(159,432)(160,431)(161,430)(162,429)(163,428)(164,427)(165,426)(166,425)(167,424)(168,423)(169,422)(170,421)(171,420)(172,419)(173,418)(174,417)(175,416)(176,415)(177,414)(178,413)(179,412)(180,411)(181,410)(182,409)(183,408)(184,407)(185,406)(186,405)(187,404)(188,403)(189,402)(190,401)(191,400)(192,399)(193,398)(194,397)(195,396)(196,395)(197,394)(198,393)(199,392)(200,391)(201,390)(202,389)(203,388)(204,387)(205,386)(206,385)(207,384)(208,383)(209,382)(210,381)(211,380)(212,379)(213,378)(214,377)(215,376)(216,375)(217,374)(218,373)(219,372)(220,371)(221,370)(222,369)(223,368)(224,367)(225,366)(226,365)(227,364)(228,363)(229,362)(230,361)(231,360)(232,359)(233,358)(234,357)(235,356)(236,355)(237,354)(238,353)(239,352)(240,351)(241,350)(242,349)(243,348)(244,347)(245,346)(246,345)(247,344)(248,343)(249,342)(250,341)(251,340)(252,339)(253,338)(254,337)(255,336)(256,335)(257,334)(258,333)(259,332)(260,331)(261,330)(262,329)(263,328)(264,327)(265,326)(266,325)(267,324)(268,323)(269,322)(270,321)(271,320)(272,319)(273,318)(274,317)(275,316)(276,315)(277,314)(278,313)(279,312)(280,311)(281,310)(282,309)(283,308)(284,307)(285,306)(286,305)(287,304)(288,303)(289,302)(290,301)(291,300)(292,299)(293,298)(294,297)(295,296));

C2×D16213 #72302129