Finite Group Databasebeta

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C3×D3845 #37528042

order23070
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3845
unambiguous semidirect productC5⋊(C3×D769)

Splittings

Fitting, socle, HallC3×C5×C769 ⋊ C2
derived, HallC5×C769 ⋊ C2×C3
HallC3×C769 ⋊ D5
SylowC769 ⋊ C3×D5
HallC3×C5 ⋊ D769
SylowC5 ⋊ C3×D769

Subgroups

subgroupG/H
center, hypercenterC3D3845
commutator, cosocle, hypocenterC5×C769C2×C3
FrattiniC1G
Fitting, socleC3×C5×C769C2
LCSG ▷ C5×C769
UCSC1 ◁ C3

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 777
transitive degreeμt(G) = 11535

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

Conjugacy classes & character table

5772 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23070, 5);