Finite Group DatabasebetaFinite Group Databasebeta| order | 36350 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D18175 |
| unambiguous semidirect product | C25⋊D727 |
| unambiguous semidirect product | C727⋊D25 |
| Fitting, derived, Hall | C25×C727 ⋊ C2 |
| Sylow | C727 ⋊ D25 |
| Sylow | C25 ⋊ D727 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C25×C727 | C2 |
| Frattini | C5 | D3635 |
| socle | C5×C727 | D5 |
| LCS | G ▷ C25×C727 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 752 |
| transitive degree | μt(G) = 18175 |
A minimal faithful permutation representation — 2 generators on 752 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,752)(28,751)(29,750)(30,749)(31,748)(32,747)(33,746)(34,745)(35,744)(36,743)(37,742)(38,741)(39,740)(40,739)(41,738)(42,737)(43,736)(44,735)(45,734)(46,733)(47,732)(48,731)(49,730)(50,729)(51,728)(52,727)(53,726)(54,725)(55,724)(56,723)(57,722)(58,721)(59,720)(60,719)(61,718)(62,717)(63,716)(64,715)(65,714)(66,713)(67,712)(68,711)(69,710)(70,709)(71,708)(72,707)(73,706)(74,705)(75,704)(76,703)(77,702)(78,701)(79,700)(80,699)(81,698)(82,697)(83,696)(84,695)(85,694)(86,693)(87,692)(88,691)(89,690)(90,689)(91,688)(92,687)(93,686)(94,685)(95,684)(96,683)(97,682)(98,681)(99,680)(100,679)(101,678)(102,677)(103,676)(104,675)(105,674)(106,673)(107,672)(108,671)(109,670)(110,669)(111,668)(112,667)(113,666)(114,665)(115,664)(116,663)(117,662)(118,661)(119,660)(120,659)(121,658)(122,657)(123,656)(124,655)(125,654)(126,653)(127,652)(128,651)(129,650)(130,649)(131,648)(132,647)(133,646)(134,645)(135,644)(136,643)(137,642)(138,641)(139,640)(140,639)(141,638)(142,637)(143,636)(144,635)(145,634)(146,633)(147,632)(148,631)(149,630)(150,629)(151,628)(152,627)(153,626)(154,625)(155,624)(156,623)(157,622)(158,621)(159,620)(160,619)(161,618)(162,617)(163,616)(164,615)(165,614)(166,613)(167,612)(168,611)(169,610)(170,609)(171,608)(172,607)(173,606)(174,605)(175,604)(176,603)(177,602)(178,601)(179,600)(180,599)(181,598)(182,597)(183,596)(184,595)(185,594)(186,593)(187,592)(188,591)(189,590)(190,589)(191,588)(192,587)(193,586)(194,585)(195,584)(196,583)(197,582)(198,581)(199,580)(200,579)(201,578)(202,577)(203,576)(204,575)(205,574)(206,573)(207,572)(208,571)(209,570)(210,569)(211,568)(212,567)(213,566)(214,565)(215,564)(216,563)(217,562)(218,561)(219,560)(220,559)(221,558)(222,557)(223,556)(224,555)(225,554)(226,553)(227,552)(228,551)(229,550)(230,549)(231,548)(232,547)(233,546)(234,545)(235,544)(236,543)(237,542)(238,541)(239,540)(240,539)(241,538)(242,537)(243,536)(244,535)(245,534)(246,533)(247,532)(248,531)(249,530)(250,529)(251,528)(252,527)(253,526)(254,525)(255,524)(256,523)(257,522)(258,521)(259,520)(260,519)(261,518)(262,517)(263,516)(264,515)(265,514)(266,513)(267,512)(268,511)(269,510)(270,509)(271,508)(272,507)(273,506)(274,505)(275,504)(276,503)(277,502)(278,501)(279,500)(280,499)(281,498)(282,497)(283,496)(284,495)(285,494)(286,493)(287,492)(288,491)(289,490)(290,489)(291,488)(292,487)(293,486)(294,485)(295,484)(296,483)(297,482)(298,481)(299,480)(300,479)(301,478)(302,477)(303,476)(304,475)(305,474)(306,473)(307,472)(308,471)(309,470)(310,469)(311,468)(312,467)(313,466)(314,465)(315,464)(316,463)(317,462)(318,461)(319,460)(320,459)(321,458)(322,457)(323,456)(324,455)(325,454)(326,453)(327,452)(328,451)(329,450)(330,449)(331,448)(332,447)(333,446)(334,445)(335,444)(336,443)(337,442)(338,441)(339,440)(340,439)(341,438)(342,437)(343,436)(344,435)(345,434)(346,433)(347,432)(348,431)(349,430)(350,429)(351,428)(352,427)(353,426)(354,425)(355,424)(356,423)(357,422)(358,421)(359,420)(360,419)(361,418)(362,417)(363,416)(364,415)(365,414)(366,413)(367,412)(368,411)(369,410)(370,409)(371,408)(372,407)(373,406)(374,405)(375,404)(376,403)(377,402)(378,401)(379,400)(380,399)(381,398)(382,397)(383,396)(384,395)(385,394)(386,393)(387,392)(388,391)(389,390)Polycyclic presentation:
2,5,727,5|b^5=d^4,b^a=b^4d,c^a=c^726,d^a=d^49089 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36350, 4);