C₂×D₁₃₈₂₃ #72290959

order	55292
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₇₆₄₆
direct product	C₂×D₁₃₈₂₃

Splittings

Fitting, socle	C₂×C₂₃×C₆₀₁ ⋊ C₂
derived, Hall	C₂₃×C₆₀₁ ⋊ C₂²
Sylow	C₆₀₁ ⋊ C₂×D₂₃
Sylow	C₂₃ ⋊ C₂×D₆₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₃₈₂₃
commutator, cosocle, hypocenter	C₂₃×C₆₀₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₂₃×C₆₀₁	C₂
LCS	G ▷ C₂₃×C₆₀₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 626
transitive degree	μ_t(G) = 27646

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

Conjugacy classes & character table

13826 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), (4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(27,626)(28,625)(29,624)(30,623)(31,622)(32,621)(33,620)(34,619)(35,618)(36,617)(37,616)(38,615)(39,614)(40,613)(41,612)(42,611)(43,610)(44,609)(45,608)(46,607)(47,606)(48,605)(49,604)(50,603)(51,602)(52,601)(53,600)(54,599)(55,598)(56,597)(57,596)(58,595)(59,594)(60,593)(61,592)(62,591)(63,590)(64,589)(65,588)(66,587)(67,586)(68,585)(69,584)(70,583)(71,582)(72,581)(73,580)(74,579)(75,578)(76,577)(77,576)(78,575)(79,574)(80,573)(81,572)(82,571)(83,570)(84,569)(85,568)(86,567)(87,566)(88,565)(89,564)(90,563)(91,562)(92,561)(93,560)(94,559)(95,558)(96,557)(97,556)(98,555)(99,554)(100,553)(101,552)(102,551)(103,550)(104,549)(105,548)(106,547)(107,546)(108,545)(109,544)(110,543)(111,542)(112,541)(113,540)(114,539)(115,538)(116,537)(117,536)(118,535)(119,534)(120,533)(121,532)(122,531)(123,530)(124,529)(125,528)(126,527)(127,526)(128,525)(129,524)(130,523)(131,522)(132,521)(133,520)(134,519)(135,518)(136,517)(137,516)(138,515)(139,514)(140,513)(141,512)(142,511)(143,510)(144,509)(145,508)(146,507)(147,506)(148,505)(149,504)(150,503)(151,502)(152,501)(153,500)(154,499)(155,498)(156,497)(157,496)(158,495)(159,494)(160,493)(161,492)(162,491)(163,490)(164,489)(165,488)(166,487)(167,486)(168,485)(169,484)(170,483)(171,482)(172,481)(173,480)(174,479)(175,478)(176,477)(177,476)(178,475)(179,474)(180,473)(181,472)(182,471)(183,470)(184,469)(185,468)(186,467)(187,466)(188,465)(189,464)(190,463)(191,462)(192,461)(193,460)(194,459)(195,458)(196,457)(197,456)(198,455)(199,454)(200,453)(201,452)(202,451)(203,450)(204,449)(205,448)(206,447)(207,446)(208,445)(209,444)(210,443)(211,442)(212,441)(213,440)(214,439)(215,438)(216,437)(217,436)(218,435)(219,434)(220,433)(221,432)(222,431)(223,430)(224,429)(225,428)(226,427)(227,426)(228,425)(229,424)(230,423)(231,422)(232,421)(233,420)(234,419)(235,418)(236,417)(237,416)(238,415)(239,414)(240,413)(241,412)(242,411)(243,410)(244,409)(245,408)(246,407)(247,406)(248,405)(249,404)(250,403)(251,402)(252,401)(253,400)(254,399)(255,398)(256,397)(257,396)(258,395)(259,394)(260,393)(261,392)(262,391)(263,390)(264,389)(265,388)(266,387)(267,386)(268,385)(269,384)(270,383)(271,382)(272,381)(273,380)(274,379)(275,378)(276,377)(277,376)(278,375)(279,374)(280,373)(281,372)(282,371)(283,370)(284,369)(285,368)(286,367)(287,366)(288,365)(289,364)(290,363)(291,362)(292,361)(293,360)(294,359)(295,358)(296,357)(297,356)(298,355)(299,354)(300,353)(301,352)(302,351)(303,350)(304,349)(305,348)(306,347)(307,346)(308,345)(309,344)(310,343)(311,342)(312,341)(313,340)(314,339)(315,338)(316,337)(317,336)(318,335)(319,334)(320,333)(321,332)(322,331)(323,330)(324,329)(325,328)(326,327));

C2×D13823 #72290959