C₂×D₁₃₁₂₉ #72289620

order	52516
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₆₂₅₈
direct product	C₂×D₁₃₁₂₉

Splittings

Fitting, socle	C₂×C₁₉×C₆₉₁ ⋊ C₂
derived, Hall	C₁₉×C₆₉₁ ⋊ C₂²
Sylow	C₆₉₁ ⋊ C₂×D₁₉
Sylow	C₁₉ ⋊ C₂×D₆₉₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₁₂₉
commutator, cosocle, hypocenter	C₁₉×C₆₉₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₉×C₆₉₁	C₂
LCS	G ▷ C₁₉×C₆₉₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 712
transitive degree	μ_t(G) = 26258

A minimal faithful permutation representation — 2 generators on 712 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712);(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(23,712)(24,711)(25,710)(26,709)(27,708)(28,707)(29,706)(30,705)(31,704)(32,703)(33,702)(34,701)(35,700)(36,699)(37,698)(38,697)(39,696)(40,695)(41,694)(42,693)(43,692)(44,691)(45,690)(46,689)(47,688)(48,687)(49,686)(50,685)(51,684)(52,683)(53,682)(54,681)(55,680)(56,679)(57,678)(58,677)(59,676)(60,675)(61,674)(62,673)(63,672)(64,671)(65,670)(66,669)(67,668)(68,667)(69,666)(70,665)(71,664)(72,663)(73,662)(74,661)(75,660)(76,659)(77,658)(78,657)(79,656)(80,655)(81,654)(82,653)(83,652)(84,651)(85,650)(86,649)(87,648)(88,647)(89,646)(90,645)(91,644)(92,643)(93,642)(94,641)(95,640)(96,639)(97,638)(98,637)(99,636)(100,635)(101,634)(102,633)(103,632)(104,631)(105,630)(106,629)(107,628)(108,627)(109,626)(110,625)(111,624)(112,623)(113,622)(114,621)(115,620)(116,619)(117,618)(118,617)(119,616)(120,615)(121,614)(122,613)(123,612)(124,611)(125,610)(126,609)(127,608)(128,607)(129,606)(130,605)(131,604)(132,603)(133,602)(134,601)(135,600)(136,599)(137,598)(138,597)(139,596)(140,595)(141,594)(142,593)(143,592)(144,591)(145,590)(146,589)(147,588)(148,587)(149,586)(150,585)(151,584)(152,583)(153,582)(154,581)(155,580)(156,579)(157,578)(158,577)(159,576)(160,575)(161,574)(162,573)(163,572)(164,571)(165,570)(166,569)(167,568)(168,567)(169,566)(170,565)(171,564)(172,563)(173,562)(174,561)(175,560)(176,559)(177,558)(178,557)(179,556)(180,555)(181,554)(182,553)(183,552)(184,551)(185,550)(186,549)(187,548)(188,547)(189,546)(190,545)(191,544)(192,543)(193,542)(194,541)(195,540)(196,539)(197,538)(198,537)(199,536)(200,535)(201,534)(202,533)(203,532)(204,531)(205,530)(206,529)(207,528)(208,527)(209,526)(210,525)(211,524)(212,523)(213,522)(214,521)(215,520)(216,519)(217,518)(218,517)(219,516)(220,515)(221,514)(222,513)(223,512)(224,511)(225,510)(226,509)(227,508)(228,507)(229,506)(230,505)(231,504)(232,503)(233,502)(234,501)(235,500)(236,499)(237,498)(238,497)(239,496)(240,495)(241,494)(242,493)(243,492)(244,491)(245,490)(246,489)(247,488)(248,487)(249,486)(250,485)(251,484)(252,483)(253,482)(254,481)(255,480)(256,479)(257,478)(258,477)(259,476)(260,475)(261,474)(262,473)(263,472)(264,471)(265,470)(266,469)(267,468)(268,467)(269,466)(270,465)(271,464)(272,463)(273,462)(274,461)(275,460)(276,459)(277,458)(278,457)(279,456)(280,455)(281,454)(282,453)(283,452)(284,451)(285,450)(286,449)(287,448)(288,447)(289,446)(290,445)(291,444)(292,443)(293,442)(294,441)(295,440)(296,439)(297,438)(298,437)(299,436)(300,435)(301,434)(302,433)(303,432)(304,431)(305,430)(306,429)(307,428)(308,427)(309,426)(310,425)(311,424)(312,423)(313,422)(314,421)(315,420)(316,419)(317,418)(318,417)(319,416)(320,415)(321,414)(322,413)(323,412)(324,411)(325,410)(326,409)(327,408)(328,407)(329,406)(330,405)(331,404)(332,403)(333,402)(334,401)(335,400)(336,399)(337,398)(338,397)(339,396)(340,395)(341,394)(342,393)(343,392)(344,391)(345,390)(346,389)(347,388)(348,387)(349,386)(350,385)(351,384)(352,383)(353,382)(354,381)(355,380)(356,379)(357,378)(358,377)(359,376)(360,375)(361,374)(362,373)(363,372)(364,371)(365,370)(366,369)(367,368)

Conjugacy classes & character table

13132 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)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712), (4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(23,712)(24,711)(25,710)(26,709)(27,708)(28,707)(29,706)(30,705)(31,704)(32,703)(33,702)(34,701)(35,700)(36,699)(37,698)(38,697)(39,696)(40,695)(41,694)(42,693)(43,692)(44,691)(45,690)(46,689)(47,688)(48,687)(49,686)(50,685)(51,684)(52,683)(53,682)(54,681)(55,680)(56,679)(57,678)(58,677)(59,676)(60,675)(61,674)(62,673)(63,672)(64,671)(65,670)(66,669)(67,668)(68,667)(69,666)(70,665)(71,664)(72,663)(73,662)(74,661)(75,660)(76,659)(77,658)(78,657)(79,656)(80,655)(81,654)(82,653)(83,652)(84,651)(85,650)(86,649)(87,648)(88,647)(89,646)(90,645)(91,644)(92,643)(93,642)(94,641)(95,640)(96,639)(97,638)(98,637)(99,636)(100,635)(101,634)(102,633)(103,632)(104,631)(105,630)(106,629)(107,628)(108,627)(109,626)(110,625)(111,624)(112,623)(113,622)(114,621)(115,620)(116,619)(117,618)(118,617)(119,616)(120,615)(121,614)(122,613)(123,612)(124,611)(125,610)(126,609)(127,608)(128,607)(129,606)(130,605)(131,604)(132,603)(133,602)(134,601)(135,600)(136,599)(137,598)(138,597)(139,596)(140,595)(141,594)(142,593)(143,592)(144,591)(145,590)(146,589)(147,588)(148,587)(149,586)(150,585)(151,584)(152,583)(153,582)(154,581)(155,580)(156,579)(157,578)(158,577)(159,576)(160,575)(161,574)(162,573)(163,572)(164,571)(165,570)(166,569)(167,568)(168,567)(169,566)(170,565)(171,564)(172,563)(173,562)(174,561)(175,560)(176,559)(177,558)(178,557)(179,556)(180,555)(181,554)(182,553)(183,552)(184,551)(185,550)(186,549)(187,548)(188,547)(189,546)(190,545)(191,544)(192,543)(193,542)(194,541)(195,540)(196,539)(197,538)(198,537)(199,536)(200,535)(201,534)(202,533)(203,532)(204,531)(205,530)(206,529)(207,528)(208,527)(209,526)(210,525)(211,524)(212,523)(213,522)(214,521)(215,520)(216,519)(217,518)(218,517)(219,516)(220,515)(221,514)(222,513)(223,512)(224,511)(225,510)(226,509)(227,508)(228,507)(229,506)(230,505)(231,504)(232,503)(233,502)(234,501)(235,500)(236,499)(237,498)(238,497)(239,496)(240,495)(241,494)(242,493)(243,492)(244,491)(245,490)(246,489)(247,488)(248,487)(249,486)(250,485)(251,484)(252,483)(253,482)(254,481)(255,480)(256,479)(257,478)(258,477)(259,476)(260,475)(261,474)(262,473)(263,472)(264,471)(265,470)(266,469)(267,468)(268,467)(269,466)(270,465)(271,464)(272,463)(273,462)(274,461)(275,460)(276,459)(277,458)(278,457)(279,456)(280,455)(281,454)(282,453)(283,452)(284,451)(285,450)(286,449)(287,448)(288,447)(289,446)(290,445)(291,444)(292,443)(293,442)(294,441)(295,440)(296,439)(297,438)(298,437)(299,436)(300,435)(301,434)(302,433)(303,432)(304,431)(305,430)(306,429)(307,428)(308,427)(309,426)(310,425)(311,424)(312,423)(313,422)(314,421)(315,420)(316,419)(317,418)(318,417)(319,416)(320,415)(321,414)(322,413)(323,412)(324,411)(325,410)(326,409)(327,408)(328,407)(329,406)(330,405)(331,404)(332,403)(333,402)(334,401)(335,400)(336,399)(337,398)(338,397)(339,396)(340,395)(341,394)(342,393)(343,392)(344,391)(345,390)(346,389)(347,388)(348,387)(349,386)(350,385)(351,384)(352,383)(353,382)(354,381)(355,380)(356,379)(357,378)(358,377)(359,376)(360,375)(361,374)(362,373)(363,372)(364,371)(365,370)(366,369)(367,368));

C2×D13129 #72289620