C₂×D₁₄₂₆₃ #72292151

order	57052
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₈₅₂₆
direct product	C₂×D₁₄₂₆₃

Splittings

Fitting, socle	C₂×C₁₇×C₈₃₉ ⋊ C₂
derived, Hall	C₁₇×C₈₃₉ ⋊ C₂²
Sylow	C₈₃₉ ⋊ C₂×D₁₇
Sylow	C₁₇ ⋊ C₂×D₈₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₄₂₆₃
commutator, cosocle, hypocenter	C₁₇×C₈₃₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₇×C₈₃₉	C₂
LCS	G ▷ C₁₇×C₈₃₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 858
transitive degree	μ_t(G) = 28526

A minimal faithful permutation representation — 2 generators on 858 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858);(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(21,858)(22,857)(23,856)(24,855)(25,854)(26,853)(27,852)(28,851)(29,850)(30,849)(31,848)(32,847)(33,846)(34,845)(35,844)(36,843)(37,842)(38,841)(39,840)(40,839)(41,838)(42,837)(43,836)(44,835)(45,834)(46,833)(47,832)(48,831)(49,830)(50,829)(51,828)(52,827)(53,826)(54,825)(55,824)(56,823)(57,822)(58,821)(59,820)(60,819)(61,818)(62,817)(63,816)(64,815)(65,814)(66,813)(67,812)(68,811)(69,810)(70,809)(71,808)(72,807)(73,806)(74,805)(75,804)(76,803)(77,802)(78,801)(79,800)(80,799)(81,798)(82,797)(83,796)(84,795)(85,794)(86,793)(87,792)(88,791)(89,790)(90,789)(91,788)(92,787)(93,786)(94,785)(95,784)(96,783)(97,782)(98,781)(99,780)(100,779)(101,778)(102,777)(103,776)(104,775)(105,774)(106,773)(107,772)(108,771)(109,770)(110,769)(111,768)(112,767)(113,766)(114,765)(115,764)(116,763)(117,762)(118,761)(119,760)(120,759)(121,758)(122,757)(123,756)(124,755)(125,754)(126,753)(127,752)(128,751)(129,750)(130,749)(131,748)(132,747)(133,746)(134,745)(135,744)(136,743)(137,742)(138,741)(139,740)(140,739)(141,738)(142,737)(143,736)(144,735)(145,734)(146,733)(147,732)(148,731)(149,730)(150,729)(151,728)(152,727)(153,726)(154,725)(155,724)(156,723)(157,722)(158,721)(159,720)(160,719)(161,718)(162,717)(163,716)(164,715)(165,714)(166,713)(167,712)(168,711)(169,710)(170,709)(171,708)(172,707)(173,706)(174,705)(175,704)(176,703)(177,702)(178,701)(179,700)(180,699)(181,698)(182,697)(183,696)(184,695)(185,694)(186,693)(187,692)(188,691)(189,690)(190,689)(191,688)(192,687)(193,686)(194,685)(195,684)(196,683)(197,682)(198,681)(199,680)(200,679)(201,678)(202,677)(203,676)(204,675)(205,674)(206,673)(207,672)(208,671)(209,670)(210,669)(211,668)(212,667)(213,666)(214,665)(215,664)(216,663)(217,662)(218,661)(219,660)(220,659)(221,658)(222,657)(223,656)(224,655)(225,654)(226,653)(227,652)(228,651)(229,650)(230,649)(231,648)(232,647)(233,646)(234,645)(235,644)(236,643)(237,642)(238,641)(239,640)(240,639)(241,638)(242,637)(243,636)(244,635)(245,634)(246,633)(247,632)(248,631)(249,630)(250,629)(251,628)(252,627)(253,626)(254,625)(255,624)(256,623)(257,622)(258,621)(259,620)(260,619)(261,618)(262,617)(263,616)(264,615)(265,614)(266,613)(267,612)(268,611)(269,610)(270,609)(271,608)(272,607)(273,606)(274,605)(275,604)(276,603)(277,602)(278,601)(279,600)(280,599)(281,598)(282,597)(283,596)(284,595)(285,594)(286,593)(287,592)(288,591)(289,590)(290,589)(291,588)(292,587)(293,586)(294,585)(295,584)(296,583)(297,582)(298,581)(299,580)(300,579)(301,578)(302,577)(303,576)(304,575)(305,574)(306,573)(307,572)(308,571)(309,570)(310,569)(311,568)(312,567)(313,566)(314,565)(315,564)(316,563)(317,562)(318,561)(319,560)(320,559)(321,558)(322,557)(323,556)(324,555)(325,554)(326,553)(327,552)(328,551)(329,550)(330,549)(331,548)(332,547)(333,546)(334,545)(335,544)(336,543)(337,542)(338,541)(339,540)(340,539)(341,538)(342,537)(343,536)(344,535)(345,534)(346,533)(347,532)(348,531)(349,530)(350,529)(351,528)(352,527)(353,526)(354,525)(355,524)(356,523)(357,522)(358,521)(359,520)(360,519)(361,518)(362,517)(363,516)(364,515)(365,514)(366,513)(367,512)(368,511)(369,510)(370,509)(371,508)(372,507)(373,506)(374,505)(375,504)(376,503)(377,502)(378,501)(379,500)(380,499)(381,498)(382,497)(383,496)(384,495)(385,494)(386,493)(387,492)(388,491)(389,490)(390,489)(391,488)(392,487)(393,486)(394,485)(395,484)(396,483)(397,482)(398,481)(399,480)(400,479)(401,478)(402,477)(403,476)(404,475)(405,474)(406,473)(407,472)(408,471)(409,470)(410,469)(411,468)(412,467)(413,466)(414,465)(415,464)(416,463)(417,462)(418,461)(419,460)(420,459)(421,458)(422,457)(423,456)(424,455)(425,454)(426,453)(427,452)(428,451)(429,450)(430,449)(431,448)(432,447)(433,446)(434,445)(435,444)(436,443)(437,442)(438,441)(439,440)

Conjugacy classes & character table

14266 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)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858), (4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(21,858)(22,857)(23,856)(24,855)(25,854)(26,853)(27,852)(28,851)(29,850)(30,849)(31,848)(32,847)(33,846)(34,845)(35,844)(36,843)(37,842)(38,841)(39,840)(40,839)(41,838)(42,837)(43,836)(44,835)(45,834)(46,833)(47,832)(48,831)(49,830)(50,829)(51,828)(52,827)(53,826)(54,825)(55,824)(56,823)(57,822)(58,821)(59,820)(60,819)(61,818)(62,817)(63,816)(64,815)(65,814)(66,813)(67,812)(68,811)(69,810)(70,809)(71,808)(72,807)(73,806)(74,805)(75,804)(76,803)(77,802)(78,801)(79,800)(80,799)(81,798)(82,797)(83,796)(84,795)(85,794)(86,793)(87,792)(88,791)(89,790)(90,789)(91,788)(92,787)(93,786)(94,785)(95,784)(96,783)(97,782)(98,781)(99,780)(100,779)(101,778)(102,777)(103,776)(104,775)(105,774)(106,773)(107,772)(108,771)(109,770)(110,769)(111,768)(112,767)(113,766)(114,765)(115,764)(116,763)(117,762)(118,761)(119,760)(120,759)(121,758)(122,757)(123,756)(124,755)(125,754)(126,753)(127,752)(128,751)(129,750)(130,749)(131,748)(132,747)(133,746)(134,745)(135,744)(136,743)(137,742)(138,741)(139,740)(140,739)(141,738)(142,737)(143,736)(144,735)(145,734)(146,733)(147,732)(148,731)(149,730)(150,729)(151,728)(152,727)(153,726)(154,725)(155,724)(156,723)(157,722)(158,721)(159,720)(160,719)(161,718)(162,717)(163,716)(164,715)(165,714)(166,713)(167,712)(168,711)(169,710)(170,709)(171,708)(172,707)(173,706)(174,705)(175,704)(176,703)(177,702)(178,701)(179,700)(180,699)(181,698)(182,697)(183,696)(184,695)(185,694)(186,693)(187,692)(188,691)(189,690)(190,689)(191,688)(192,687)(193,686)(194,685)(195,684)(196,683)(197,682)(198,681)(199,680)(200,679)(201,678)(202,677)(203,676)(204,675)(205,674)(206,673)(207,672)(208,671)(209,670)(210,669)(211,668)(212,667)(213,666)(214,665)(215,664)(216,663)(217,662)(218,661)(219,660)(220,659)(221,658)(222,657)(223,656)(224,655)(225,654)(226,653)(227,652)(228,651)(229,650)(230,649)(231,648)(232,647)(233,646)(234,645)(235,644)(236,643)(237,642)(238,641)(239,640)(240,639)(241,638)(242,637)(243,636)(244,635)(245,634)(246,633)(247,632)(248,631)(249,630)(250,629)(251,628)(252,627)(253,626)(254,625)(255,624)(256,623)(257,622)(258,621)(259,620)(260,619)(261,618)(262,617)(263,616)(264,615)(265,614)(266,613)(267,612)(268,611)(269,610)(270,609)(271,608)(272,607)(273,606)(274,605)(275,604)(276,603)(277,602)(278,601)(279,600)(280,599)(281,598)(282,597)(283,596)(284,595)(285,594)(286,593)(287,592)(288,591)(289,590)(290,589)(291,588)(292,587)(293,586)(294,585)(295,584)(296,583)(297,582)(298,581)(299,580)(300,579)(301,578)(302,577)(303,576)(304,575)(305,574)(306,573)(307,572)(308,571)(309,570)(310,569)(311,568)(312,567)(313,566)(314,565)(315,564)(316,563)(317,562)(318,561)(319,560)(320,559)(321,558)(322,557)(323,556)(324,555)(325,554)(326,553)(327,552)(328,551)(329,550)(330,549)(331,548)(332,547)(333,546)(334,545)(335,544)(336,543)(337,542)(338,541)(339,540)(340,539)(341,538)(342,537)(343,536)(344,535)(345,534)(346,533)(347,532)(348,531)(349,530)(350,529)(351,528)(352,527)(353,526)(354,525)(355,524)(356,523)(357,522)(358,521)(359,520)(360,519)(361,518)(362,517)(363,516)(364,515)(365,514)(366,513)(367,512)(368,511)(369,510)(370,509)(371,508)(372,507)(373,506)(374,505)(375,504)(376,503)(377,502)(378,501)(379,500)(380,499)(381,498)(382,497)(383,496)(384,495)(385,494)(386,493)(387,492)(388,491)(389,490)(390,489)(391,488)(392,487)(393,486)(394,485)(395,484)(396,483)(397,482)(398,481)(399,480)(400,479)(401,478)(402,477)(403,476)(404,475)(405,474)(406,473)(407,472)(408,471)(409,470)(410,469)(411,468)(412,467)(413,466)(414,465)(415,464)(416,463)(417,462)(418,461)(419,460)(420,459)(421,458)(422,457)(423,456)(424,455)(425,454)(426,453)(427,452)(428,451)(429,450)(430,449)(431,448)(432,447)(433,446)(434,445)(435,444)(436,443)(437,442)(438,441)(439,440));

C2×D14263 #72292151