Finite Group DatabasebetaFinite Group Databasebeta| order | 26816 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic6704 |
| Sylow | C419 ⋊ Q64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D6704 |
| commutator, cosocle | C16×C419 | C22 |
| Frattini | C16 | C2×D419 |
| Fitting | C32×C419 | C2 |
| socle, prime radical | C2×C419 | D16 |
| hypercenter | C32 | D419 |
| hypocenter | C419 | Q64 |
| LCS | G ▷ C16×C419 ▷ C8×C419 ▷ C4×C419 ▷ C2×C419 ▷ C419 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 483 |
| transitive degree | μt(G) = 26816 |
A minimal faithful permutation representation — 2 generators on 483 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);(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)(66,483)(67,482)(68,481)(69,480)(70,479)(71,478)(72,477)(73,476)(74,475)(75,474)(76,473)(77,472)(78,471)(79,470)(80,469)(81,468)(82,467)(83,466)(84,465)(85,464)(86,463)(87,462)(88,461)(89,460)(90,459)(91,458)(92,457)(93,456)(94,455)(95,454)(96,453)(97,452)(98,451)(99,450)(100,449)(101,448)(102,447)(103,446)(104,445)(105,444)(106,443)(107,442)(108,441)(109,440)(110,439)(111,438)(112,437)(113,436)(114,435)(115,434)(116,433)(117,432)(118,431)(119,430)(120,429)(121,428)(122,427)(123,426)(124,425)(125,424)(126,423)(127,422)(128,421)(129,420)(130,419)(131,418)(132,417)(133,416)(134,415)(135,414)(136,413)(137,412)(138,411)(139,410)(140,409)(141,408)(142,407)(143,406)(144,405)(145,404)(146,403)(147,402)(148,401)(149,400)(150,399)(151,398)(152,397)(153,396)(154,395)(155,394)(156,393)(157,392)(158,391)(159,390)(160,389)(161,388)(162,387)(163,386)(164,385)(165,384)(166,383)(167,382)(168,381)(169,380)(170,379)(171,378)(172,377)(173,376)(174,375)(175,374)(176,373)(177,372)(178,371)(179,370)(180,369)(181,368)(182,367)(183,366)(184,365)(185,364)(186,363)(187,362)(188,361)(189,360)(190,359)(191,358)(192,357)(193,356)(194,355)(195,354)(196,353)(197,352)(198,351)(199,350)(200,349)(201,348)(202,347)(203,346)(204,345)(205,344)(206,343)(207,342)(208,341)(209,340)(210,339)(211,338)(212,337)(213,336)(214,335)(215,334)(216,333)(217,332)(218,331)(219,330)(220,329)(221,328)(222,327)(223,326)(224,325)(225,324)(226,323)(227,322)(228,321)(229,320)(230,319)(231,318)(232,317)(233,316)(234,315)(235,314)(236,313)(237,312)(238,311)(239,310)(240,309)(241,308)(242,307)(243,306)(244,305)(245,304)(246,303)(247,302)(248,301)(249,300)(250,299)(251,298)(252,297)(253,296)(254,295)(255,294)(256,293)(257,292)(258,291)(259,290)(260,289)(261,288)(262,287)(263,286)(264,285)(265,284)(266,283)(267,282)(268,281)(269,280)(270,279)(271,278)(272,277)(273,276)(274,275)Polycyclic presentation:
2^6,419|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^418,c^b=cd,d^b=de,e^b=ef,g^b=g^4186707 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26816, 390);