C₂×D₁₂₆₂₃ #72288424

order	50492
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₅₂₄₆
direct product	C₂×D₁₂₆₂₃

Splittings

Fitting, socle	C₂×C₁₃×C₉₇₁ ⋊ C₂
derived, Hall	C₁₃×C₉₇₁ ⋊ C₂²
Sylow	C₉₇₁ ⋊ C₂×D₁₃
Sylow	C₁₃ ⋊ C₂×D₉₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₂₆₂₃
commutator, cosocle, hypocenter	C₁₃×C₉₇₁	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₁₃×C₉₇₁	C₂
LCS	G ▷ C₁₃×C₉₇₁
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 986
transitive degree	μ_t(G) = 25246

A minimal faithful permutation representation — 2 generators on 986 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986);(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(17,986)(18,985)(19,984)(20,983)(21,982)(22,981)(23,980)(24,979)(25,978)(26,977)(27,976)(28,975)(29,974)(30,973)(31,972)(32,971)(33,970)(34,969)(35,968)(36,967)(37,966)(38,965)(39,964)(40,963)(41,962)(42,961)(43,960)(44,959)(45,958)(46,957)(47,956)(48,955)(49,954)(50,953)(51,952)(52,951)(53,950)(54,949)(55,948)(56,947)(57,946)(58,945)(59,944)(60,943)(61,942)(62,941)(63,940)(64,939)(65,938)(66,937)(67,936)(68,935)(69,934)(70,933)(71,932)(72,931)(73,930)(74,929)(75,928)(76,927)(77,926)(78,925)(79,924)(80,923)(81,922)(82,921)(83,920)(84,919)(85,918)(86,917)(87,916)(88,915)(89,914)(90,913)(91,912)(92,911)(93,910)(94,909)(95,908)(96,907)(97,906)(98,905)(99,904)(100,903)(101,902)(102,901)(103,900)(104,899)(105,898)(106,897)(107,896)(108,895)(109,894)(110,893)(111,892)(112,891)(113,890)(114,889)(115,888)(116,887)(117,886)(118,885)(119,884)(120,883)(121,882)(122,881)(123,880)(124,879)(125,878)(126,877)(127,876)(128,875)(129,874)(130,873)(131,872)(132,871)(133,870)(134,869)(135,868)(136,867)(137,866)(138,865)(139,864)(140,863)(141,862)(142,861)(143,860)(144,859)(145,858)(146,857)(147,856)(148,855)(149,854)(150,853)(151,852)(152,851)(153,850)(154,849)(155,848)(156,847)(157,846)(158,845)(159,844)(160,843)(161,842)(162,841)(163,840)(164,839)(165,838)(166,837)(167,836)(168,835)(169,834)(170,833)(171,832)(172,831)(173,830)(174,829)(175,828)(176,827)(177,826)(178,825)(179,824)(180,823)(181,822)(182,821)(183,820)(184,819)(185,818)(186,817)(187,816)(188,815)(189,814)(190,813)(191,812)(192,811)(193,810)(194,809)(195,808)(196,807)(197,806)(198,805)(199,804)(200,803)(201,802)(202,801)(203,800)(204,799)(205,798)(206,797)(207,796)(208,795)(209,794)(210,793)(211,792)(212,791)(213,790)(214,789)(215,788)(216,787)(217,786)(218,785)(219,784)(220,783)(221,782)(222,781)(223,780)(224,779)(225,778)(226,777)(227,776)(228,775)(229,774)(230,773)(231,772)(232,771)(233,770)(234,769)(235,768)(236,767)(237,766)(238,765)(239,764)(240,763)(241,762)(242,761)(243,760)(244,759)(245,758)(246,757)(247,756)(248,755)(249,754)(250,753)(251,752)(252,751)(253,750)(254,749)(255,748)(256,747)(257,746)(258,745)(259,744)(260,743)(261,742)(262,741)(263,740)(264,739)(265,738)(266,737)(267,736)(268,735)(269,734)(270,733)(271,732)(272,731)(273,730)(274,729)(275,728)(276,727)(277,726)(278,725)(279,724)(280,723)(281,722)(282,721)(283,720)(284,719)(285,718)(286,717)(287,716)(288,715)(289,714)(290,713)(291,712)(292,711)(293,710)(294,709)(295,708)(296,707)(297,706)(298,705)(299,704)(300,703)(301,702)(302,701)(303,700)(304,699)(305,698)(306,697)(307,696)(308,695)(309,694)(310,693)(311,692)(312,691)(313,690)(314,689)(315,688)(316,687)(317,686)(318,685)(319,684)(320,683)(321,682)(322,681)(323,680)(324,679)(325,678)(326,677)(327,676)(328,675)(329,674)(330,673)(331,672)(332,671)(333,670)(334,669)(335,668)(336,667)(337,666)(338,665)(339,664)(340,663)(341,662)(342,661)(343,660)(344,659)(345,658)(346,657)(347,656)(348,655)(349,654)(350,653)(351,652)(352,651)(353,650)(354,649)(355,648)(356,647)(357,646)(358,645)(359,644)(360,643)(361,642)(362,641)(363,640)(364,639)(365,638)(366,637)(367,636)(368,635)(369,634)(370,633)(371,632)(372,631)(373,630)(374,629)(375,628)(376,627)(377,626)(378,625)(379,624)(380,623)(381,622)(382,621)(383,620)(384,619)(385,618)(386,617)(387,616)(388,615)(389,614)(390,613)(391,612)(392,611)(393,610)(394,609)(395,608)(396,607)(397,606)(398,605)(399,604)(400,603)(401,602)(402,601)(403,600)(404,599)(405,598)(406,597)(407,596)(408,595)(409,594)(410,593)(411,592)(412,591)(413,590)(414,589)(415,588)(416,587)(417,586)(418,585)(419,584)(420,583)(421,582)(422,581)(423,580)(424,579)(425,578)(426,577)(427,576)(428,575)(429,574)(430,573)(431,572)(432,571)(433,570)(434,569)(435,568)(436,567)(437,566)(438,565)(439,564)(440,563)(441,562)(442,561)(443,560)(444,559)(445,558)(446,557)(447,556)(448,555)(449,554)(450,553)(451,552)(452,551)(453,550)(454,549)(455,548)(456,547)(457,546)(458,545)(459,544)(460,543)(461,542)(462,541)(463,540)(464,539)(465,538)(466,537)(467,536)(468,535)(469,534)(470,533)(471,532)(472,531)(473,530)(474,529)(475,528)(476,527)(477,526)(478,525)(479,524)(480,523)(481,522)(482,521)(483,520)(484,519)(485,518)(486,517)(487,516)(488,515)(489,514)(490,513)(491,512)(492,511)(493,510)(494,509)(495,508)(496,507)(497,506)(498,505)(499,504)(500,503)(501,502)

Conjugacy classes & character table

12626 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)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986), (4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(17,986)(18,985)(19,984)(20,983)(21,982)(22,981)(23,980)(24,979)(25,978)(26,977)(27,976)(28,975)(29,974)(30,973)(31,972)(32,971)(33,970)(34,969)(35,968)(36,967)(37,966)(38,965)(39,964)(40,963)(41,962)(42,961)(43,960)(44,959)(45,958)(46,957)(47,956)(48,955)(49,954)(50,953)(51,952)(52,951)(53,950)(54,949)(55,948)(56,947)(57,946)(58,945)(59,944)(60,943)(61,942)(62,941)(63,940)(64,939)(65,938)(66,937)(67,936)(68,935)(69,934)(70,933)(71,932)(72,931)(73,930)(74,929)(75,928)(76,927)(77,926)(78,925)(79,924)(80,923)(81,922)(82,921)(83,920)(84,919)(85,918)(86,917)(87,916)(88,915)(89,914)(90,913)(91,912)(92,911)(93,910)(94,909)(95,908)(96,907)(97,906)(98,905)(99,904)(100,903)(101,902)(102,901)(103,900)(104,899)(105,898)(106,897)(107,896)(108,895)(109,894)(110,893)(111,892)(112,891)(113,890)(114,889)(115,888)(116,887)(117,886)(118,885)(119,884)(120,883)(121,882)(122,881)(123,880)(124,879)(125,878)(126,877)(127,876)(128,875)(129,874)(130,873)(131,872)(132,871)(133,870)(134,869)(135,868)(136,867)(137,866)(138,865)(139,864)(140,863)(141,862)(142,861)(143,860)(144,859)(145,858)(146,857)(147,856)(148,855)(149,854)(150,853)(151,852)(152,851)(153,850)(154,849)(155,848)(156,847)(157,846)(158,845)(159,844)(160,843)(161,842)(162,841)(163,840)(164,839)(165,838)(166,837)(167,836)(168,835)(169,834)(170,833)(171,832)(172,831)(173,830)(174,829)(175,828)(176,827)(177,826)(178,825)(179,824)(180,823)(181,822)(182,821)(183,820)(184,819)(185,818)(186,817)(187,816)(188,815)(189,814)(190,813)(191,812)(192,811)(193,810)(194,809)(195,808)(196,807)(197,806)(198,805)(199,804)(200,803)(201,802)(202,801)(203,800)(204,799)(205,798)(206,797)(207,796)(208,795)(209,794)(210,793)(211,792)(212,791)(213,790)(214,789)(215,788)(216,787)(217,786)(218,785)(219,784)(220,783)(221,782)(222,781)(223,780)(224,779)(225,778)(226,777)(227,776)(228,775)(229,774)(230,773)(231,772)(232,771)(233,770)(234,769)(235,768)(236,767)(237,766)(238,765)(239,764)(240,763)(241,762)(242,761)(243,760)(244,759)(245,758)(246,757)(247,756)(248,755)(249,754)(250,753)(251,752)(252,751)(253,750)(254,749)(255,748)(256,747)(257,746)(258,745)(259,744)(260,743)(261,742)(262,741)(263,740)(264,739)(265,738)(266,737)(267,736)(268,735)(269,734)(270,733)(271,732)(272,731)(273,730)(274,729)(275,728)(276,727)(277,726)(278,725)(279,724)(280,723)(281,722)(282,721)(283,720)(284,719)(285,718)(286,717)(287,716)(288,715)(289,714)(290,713)(291,712)(292,711)(293,710)(294,709)(295,708)(296,707)(297,706)(298,705)(299,704)(300,703)(301,702)(302,701)(303,700)(304,699)(305,698)(306,697)(307,696)(308,695)(309,694)(310,693)(311,692)(312,691)(313,690)(314,689)(315,688)(316,687)(317,686)(318,685)(319,684)(320,683)(321,682)(322,681)(323,680)(324,679)(325,678)(326,677)(327,676)(328,675)(329,674)(330,673)(331,672)(332,671)(333,670)(334,669)(335,668)(336,667)(337,666)(338,665)(339,664)(340,663)(341,662)(342,661)(343,660)(344,659)(345,658)(346,657)(347,656)(348,655)(349,654)(350,653)(351,652)(352,651)(353,650)(354,649)(355,648)(356,647)(357,646)(358,645)(359,644)(360,643)(361,642)(362,641)(363,640)(364,639)(365,638)(366,637)(367,636)(368,635)(369,634)(370,633)(371,632)(372,631)(373,630)(374,629)(375,628)(376,627)(377,626)(378,625)(379,624)(380,623)(381,622)(382,621)(383,620)(384,619)(385,618)(386,617)(387,616)(388,615)(389,614)(390,613)(391,612)(392,611)(393,610)(394,609)(395,608)(396,607)(397,606)(398,605)(399,604)(400,603)(401,602)(402,601)(403,600)(404,599)(405,598)(406,597)(407,596)(408,595)(409,594)(410,593)(411,592)(412,591)(413,590)(414,589)(415,588)(416,587)(417,586)(418,585)(419,584)(420,583)(421,582)(422,581)(423,580)(424,579)(425,578)(426,577)(427,576)(428,575)(429,574)(430,573)(431,572)(432,571)(433,570)(434,569)(435,568)(436,567)(437,566)(438,565)(439,564)(440,563)(441,562)(442,561)(443,560)(444,559)(445,558)(446,557)(447,556)(448,555)(449,554)(450,553)(451,552)(452,551)(453,550)(454,549)(455,548)(456,547)(457,546)(458,545)(459,544)(460,543)(461,542)(462,541)(463,540)(464,539)(465,538)(466,537)(467,536)(468,535)(469,534)(470,533)(471,532)(472,531)(473,530)(474,529)(475,528)(476,527)(477,526)(478,525)(479,524)(480,523)(481,522)(482,521)(483,520)(484,519)(485,518)(486,517)(487,516)(488,515)(489,514)(490,513)(491,512)(492,511)(493,510)(494,509)(495,508)(496,507)(497,506)(498,505)(499,504)(500,503)(501,502));

C2×D12623 #72288424