Finite Group DatabasebetaFinite Group Databasebeta| order | 31936 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic7984 |
| Sylow | C499 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D7984 |
| commutator, cosocle | C16×C499 | C22 |
| Frattini | C16 | C2×D499 |
| Fitting | C32×C499 | C2 |
| socle, prime radical | C2×C499 | D16 |
| hypercenter | C32 | D499 |
| hypocenter | C499 | Q64 |
| LCS | G ▷ C16×C499 ▷ C8×C499 ▷ C4×C499 ▷ C2×C499 ▷ C499 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 563 |
| transitive degree | μt(G) = 31936 |
A minimal faithful permutation representation — 2 generators on 563 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(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);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,563)(67,562)(68,561)(69,560)(70,559)(71,558)(72,557)(73,556)(74,555)(75,554)(76,553)(77,552)(78,551)(79,550)(80,549)(81,548)(82,547)(83,546)(84,545)(85,544)(86,543)(87,542)(88,541)(89,540)(90,539)(91,538)(92,537)(93,536)(94,535)(95,534)(96,533)(97,532)(98,531)(99,530)(100,529)(101,528)(102,527)(103,526)(104,525)(105,524)(106,523)(107,522)(108,521)(109,520)(110,519)(111,518)(112,517)(113,516)(114,515)(115,514)(116,513)(117,512)(118,511)(119,510)(120,509)(121,508)(122,507)(123,506)(124,505)(125,504)(126,503)(127,502)(128,501)(129,500)(130,499)(131,498)(132,497)(133,496)(134,495)(135,494)(136,493)(137,492)(138,491)(139,490)(140,489)(141,488)(142,487)(143,486)(144,485)(145,484)(146,483)(147,482)(148,481)(149,480)(150,479)(151,478)(152,477)(153,476)(154,475)(155,474)(156,473)(157,472)(158,471)(159,470)(160,469)(161,468)(162,467)(163,466)(164,465)(165,464)(166,463)(167,462)(168,461)(169,460)(170,459)(171,458)(172,457)(173,456)(174,455)(175,454)(176,453)(177,452)(178,451)(179,450)(180,449)(181,448)(182,447)(183,446)(184,445)(185,444)(186,443)(187,442)(188,441)(189,440)(190,439)(191,438)(192,437)(193,436)(194,435)(195,434)(196,433)(197,432)(198,431)(199,430)(200,429)(201,428)(202,427)(203,426)(204,425)(205,424)(206,423)(207,422)(208,421)(209,420)(210,419)(211,418)(212,417)(213,416)(214,415)(215,414)(216,413)(217,412)(218,411)(219,410)(220,409)(221,408)(222,407)(223,406)(224,405)(225,404)(226,403)(227,402)(228,401)(229,400)(230,399)(231,398)(232,397)(233,396)(234,395)(235,394)(236,393)(237,392)(238,391)(239,390)(240,389)(241,388)(242,387)(243,386)(244,385)(245,384)(246,383)(247,382)(248,381)(249,380)(250,379)(251,378)(252,377)(253,376)(254,375)(255,374)(256,373)(257,372)(258,371)(259,370)(260,369)(261,368)(262,367)(263,366)(264,365)(265,364)(266,363)(267,362)(268,361)(269,360)(270,359)(271,358)(272,357)(273,356)(274,355)(275,354)(276,353)(277,352)(278,351)(279,350)(280,349)(281,348)(282,347)(283,346)(284,345)(285,344)(286,343)(287,342)(288,341)(289,340)(290,339)(291,338)(292,337)(293,336)(294,335)(295,334)(296,333)(297,332)(298,331)(299,330)(300,329)(301,328)(302,327)(303,326)(304,325)(305,324)(306,323)(307,322)(308,321)(309,320)(310,319)(311,318)(312,317)(313,316)(314,315)Polycyclic presentation:
2^6,499|a^2=f,b^2=f,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^498,c^b=cd,d^b=de,e^b=ef,g^b=g^4987987 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31936, 390);