Finite Group Databasebeta

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C2×D16149 #72302017

order64596
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD32298
direct productC2×D16149

Splittings

Fitting, socleC2×C3×C7×C769 ⋊ C2
derived, HallC3×C7×C769 ⋊ C22
HallC7×C769 ⋊ C2×D3
HallC3×C769 ⋊ C2×D7
SylowC769 ⋊ C2×D21
HallC3×C7 ⋊ C2×D769
SylowC7 ⋊ C2×D2307
SylowC3 ⋊ C2×D5383

Subgroups

subgroupG/H
center, hypercenterC2D16149
commutator, cosocle, hypocenterC3×C7×C769C22
FrattiniC1G
Fitting, socleC2×C3×C7×C769C2
LCSG ▷ C3×C7×C769
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 781
transitive degreeμt(G) = 32298

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

Conjugacy classes & character table

16152 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), (4,5)(7,12)(8,11)(9,10)(14,781)(15,780)(16,779)(17,778)(18,777)(19,776)(20,775)(21,774)(22,773)(23,772)(24,771)(25,770)(26,769)(27,768)(28,767)(29,766)(30,765)(31,764)(32,763)(33,762)(34,761)(35,760)(36,759)(37,758)(38,757)(39,756)(40,755)(41,754)(42,753)(43,752)(44,751)(45,750)(46,749)(47,748)(48,747)(49,746)(50,745)(51,744)(52,743)(53,742)(54,741)(55,740)(56,739)(57,738)(58,737)(59,736)(60,735)(61,734)(62,733)(63,732)(64,731)(65,730)(66,729)(67,728)(68,727)(69,726)(70,725)(71,724)(72,723)(73,722)(74,721)(75,720)(76,719)(77,718)(78,717)(79,716)(80,715)(81,714)(82,713)(83,712)(84,711)(85,710)(86,709)(87,708)(88,707)(89,706)(90,705)(91,704)(92,703)(93,702)(94,701)(95,700)(96,699)(97,698)(98,697)(99,696)(100,695)(101,694)(102,693)(103,692)(104,691)(105,690)(106,689)(107,688)(108,687)(109,686)(110,685)(111,684)(112,683)(113,682)(114,681)(115,680)(116,679)(117,678)(118,677)(119,676)(120,675)(121,674)(122,673)(123,672)(124,671)(125,670)(126,669)(127,668)(128,667)(129,666)(130,665)(131,664)(132,663)(133,662)(134,661)(135,660)(136,659)(137,658)(138,657)(139,656)(140,655)(141,654)(142,653)(143,652)(144,651)(145,650)(146,649)(147,648)(148,647)(149,646)(150,645)(151,644)(152,643)(153,642)(154,641)(155,640)(156,639)(157,638)(158,637)(159,636)(160,635)(161,634)(162,633)(163,632)(164,631)(165,630)(166,629)(167,628)(168,627)(169,626)(170,625)(171,624)(172,623)(173,622)(174,621)(175,620)(176,619)(177,618)(178,617)(179,616)(180,615)(181,614)(182,613)(183,612)(184,611)(185,610)(186,609)(187,608)(188,607)(189,606)(190,605)(191,604)(192,603)(193,602)(194,601)(195,600)(196,599)(197,598)(198,597)(199,596)(200,595)(201,594)(202,593)(203,592)(204,591)(205,590)(206,589)(207,588)(208,587)(209,586)(210,585)(211,584)(212,583)(213,582)(214,581)(215,580)(216,579)(217,578)(218,577)(219,576)(220,575)(221,574)(222,573)(223,572)(224,571)(225,570)(226,569)(227,568)(228,567)(229,566)(230,565)(231,564)(232,563)(233,562)(234,561)(235,560)(236,559)(237,558)(238,557)(239,556)(240,555)(241,554)(242,553)(243,552)(244,551)(245,550)(246,549)(247,548)(248,547)(249,546)(250,545)(251,544)(252,543)(253,542)(254,541)(255,540)(256,539)(257,538)(258,537)(259,536)(260,535)(261,534)(262,533)(263,532)(264,531)(265,530)(266,529)(267,528)(268,527)(269,526)(270,525)(271,524)(272,523)(273,522)(274,521)(275,520)(276,519)(277,518)(278,517)(279,516)(280,515)(281,514)(282,513)(283,512)(284,511)(285,510)(286,509)(287,508)(288,507)(289,506)(290,505)(291,504)(292,503)(293,502)(294,501)(295,500)(296,499)(297,498)(298,497)(299,496)(300,495)(301,494)(302,493)(303,492)(304,491)(305,490)(306,489)(307,488)(308,487)(309,486)(310,485)(311,484)(312,483)(313,482)(314,481)(315,480)(316,479)(317,478)(318,477)(319,476)(320,475)(321,474)(322,473)(323,472)(324,471)(325,470)(326,469)(327,468)(328,467)(329,466)(330,465)(331,464)(332,463)(333,462)(334,461)(335,460)(336,459)(337,458)(338,457)(339,456)(340,455)(341,454)(342,453)(343,452)(344,451)(345,450)(346,449)(347,448)(348,447)(349,446)(350,445)(351,444)(352,443)(353,442)(354,441)(355,440)(356,439)(357,438)(358,437)(359,436)(360,435)(361,434)(362,433)(363,432)(364,431)(365,430)(366,429)(367,428)(368,427)(369,426)(370,425)(371,424)(372,423)(373,422)(374,421)(375,420)(376,419)(377,418)(378,417)(379,416)(380,415)(381,414)(382,413)(383,412)(384,411)(385,410)(386,409)(387,408)(388,407)(389,406)(390,405)(391,404)(392,403)(393,402)(394,401)(395,400)(396,399)(397,398));