Finite Group DatabasebetaFinite Group Databasebeta| order | 23328 |
| automorphism | Aut(G) order 45349632 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D11664 |
| Fitting | C16×C729 ⋊ C2 |
| Sylow | C729 ⋊ D16 |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 745 |
| transitive degree | μt(G) = 11664 |
A minimal faithful permutation representation — 2 generators on 745 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,745)(19,744)(20,743)(21,742)(22,741)(23,740)(24,739)(25,738)(26,737)(27,736)(28,735)(29,734)(30,733)(31,732)(32,731)(33,730)(34,729)(35,728)(36,727)(37,726)(38,725)(39,724)(40,723)(41,722)(42,721)(43,720)(44,719)(45,718)(46,717)(47,716)(48,715)(49,714)(50,713)(51,712)(52,711)(53,710)(54,709)(55,708)(56,707)(57,706)(58,705)(59,704)(60,703)(61,702)(62,701)(63,700)(64,699)(65,698)(66,697)(67,696)(68,695)(69,694)(70,693)(71,692)(72,691)(73,690)(74,689)(75,688)(76,687)(77,686)(78,685)(79,684)(80,683)(81,682)(82,681)(83,680)(84,679)(85,678)(86,677)(87,676)(88,675)(89,674)(90,673)(91,672)(92,671)(93,670)(94,669)(95,668)(96,667)(97,666)(98,665)(99,664)(100,663)(101,662)(102,661)(103,660)(104,659)(105,658)(106,657)(107,656)(108,655)(109,654)(110,653)(111,652)(112,651)(113,650)(114,649)(115,648)(116,647)(117,646)(118,645)(119,644)(120,643)(121,642)(122,641)(123,640)(124,639)(125,638)(126,637)(127,636)(128,635)(129,634)(130,633)(131,632)(132,631)(133,630)(134,629)(135,628)(136,627)(137,626)(138,625)(139,624)(140,623)(141,622)(142,621)(143,620)(144,619)(145,618)(146,617)(147,616)(148,615)(149,614)(150,613)(151,612)(152,611)(153,610)(154,609)(155,608)(156,607)(157,606)(158,605)(159,604)(160,603)(161,602)(162,601)(163,600)(164,599)(165,598)(166,597)(167,596)(168,595)(169,594)(170,593)(171,592)(172,591)(173,590)(174,589)(175,588)(176,587)(177,586)(178,585)(179,584)(180,583)(181,582)(182,581)(183,580)(184,579)(185,578)(186,577)(187,576)(188,575)(189,574)(190,573)(191,572)(192,571)(193,570)(194,569)(195,568)(196,567)(197,566)(198,565)(199,564)(200,563)(201,562)(202,561)(203,560)(204,559)(205,558)(206,557)(207,556)(208,555)(209,554)(210,553)(211,552)(212,551)(213,550)(214,549)(215,548)(216,547)(217,546)(218,545)(219,544)(220,543)(221,542)(222,541)(223,540)(224,539)(225,538)(226,537)(227,536)(228,535)(229,534)(230,533)(231,532)(232,531)(233,530)(234,529)(235,528)(236,527)(237,526)(238,525)(239,524)(240,523)(241,522)(242,521)(243,520)(244,519)(245,518)(246,517)(247,516)(248,515)(249,514)(250,513)(251,512)(252,511)(253,510)(254,509)(255,508)(256,507)(257,506)(258,505)(259,504)(260,503)(261,502)(262,501)(263,500)(264,499)(265,498)(266,497)(267,496)(268,495)(269,494)(270,493)(271,492)(272,491)(273,490)(274,489)(275,488)(276,487)(277,486)(278,485)(279,484)(280,483)(281,482)(282,481)(283,480)(284,479)(285,478)(286,477)(287,476)(288,475)(289,474)(290,473)(291,472)(292,471)(293,470)(294,469)(295,468)(296,467)(297,466)(298,465)(299,464)(300,463)(301,462)(302,461)(303,460)(304,459)(305,458)(306,457)(307,456)(308,455)(309,454)(310,453)(311,452)(312,451)(313,450)(314,449)(315,448)(316,447)(317,446)(318,445)(319,444)(320,443)(321,442)(322,441)(323,440)(324,439)(325,438)(326,437)(327,436)(328,435)(329,434)(330,433)(331,432)(332,431)(333,430)(334,429)(335,428)(336,427)(337,426)(338,425)(339,424)(340,423)(341,422)(342,421)(343,420)(344,419)(345,418)(346,417)(347,416)(348,415)(349,414)(350,413)(351,412)(352,411)(353,410)(354,409)(355,408)(356,407)(357,406)(358,405)(359,404)(360,403)(361,402)(362,401)(363,400)(364,399)(365,398)(366,397)(367,396)(368,395)(369,394)(370,393)(371,392)(372,391)(373,390)(374,389)(375,388)(376,387)(377,386)(378,385)(379,384)(380,383)(381,382)Polycyclic presentation:
2^5,3^6|b^2=cd,c^2=de,d^2=e,f^3=g,g^3=h,h^3=i,i^3=j,j^3=k,b^a=bc,c^a=cd,d^a=de,f^a=f^2g^2h^2i^2j^2k^2,g^a=g^2h^2i^2j^2k^2,h^a=h^2i^2j^2k^2,i^a=i^2j^2k^2,j^a=j^2k^2,k^a=k^25835 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D23328 |
| maximal normal subgroup of | D23328 |
| maximal quotient of | Dic11664, D23328 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2*F.8^2*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 10, 1, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);