Finite Group DatabasebetaFinite Group Databasebeta| order | 26816 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D13408 |
| Fitting | C32×C419 ⋊ C2 |
| Sylow | C419 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D6704 |
| commutator, cosocle | C16×C419 | C22 |
| Frattini | C16 | C2×D419 |
| Fitting | C32×C419 | C2 |
| socle | C2×C419 | D16 |
| hypercenter | C32 | D419 |
| hypocenter | C419 | D32 |
| LCS | G ▷ C16×C419 ▷ C8×C419 ▷ C4×C419 ▷ C2×C419 ▷ C419 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 451 |
| transitive degree | μt(G) = 13408 |
A minimal faithful permutation representation — 2 generators on 451 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);(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,451)(35,450)(36,449)(37,448)(38,447)(39,446)(40,445)(41,444)(42,443)(43,442)(44,441)(45,440)(46,439)(47,438)(48,437)(49,436)(50,435)(51,434)(52,433)(53,432)(54,431)(55,430)(56,429)(57,428)(58,427)(59,426)(60,425)(61,424)(62,423)(63,422)(64,421)(65,420)(66,419)(67,418)(68,417)(69,416)(70,415)(71,414)(72,413)(73,412)(74,411)(75,410)(76,409)(77,408)(78,407)(79,406)(80,405)(81,404)(82,403)(83,402)(84,401)(85,400)(86,399)(87,398)(88,397)(89,396)(90,395)(91,394)(92,393)(93,392)(94,391)(95,390)(96,389)(97,388)(98,387)(99,386)(100,385)(101,384)(102,383)(103,382)(104,381)(105,380)(106,379)(107,378)(108,377)(109,376)(110,375)(111,374)(112,373)(113,372)(114,371)(115,370)(116,369)(117,368)(118,367)(119,366)(120,365)(121,364)(122,363)(123,362)(124,361)(125,360)(126,359)(127,358)(128,357)(129,356)(130,355)(131,354)(132,353)(133,352)(134,351)(135,350)(136,349)(137,348)(138,347)(139,346)(140,345)(141,344)(142,343)(143,342)(144,341)(145,340)(146,339)(147,338)(148,337)(149,336)(150,335)(151,334)(152,333)(153,332)(154,331)(155,330)(156,329)(157,328)(158,327)(159,326)(160,325)(161,324)(162,323)(163,322)(164,321)(165,320)(166,319)(167,318)(168,317)(169,316)(170,315)(171,314)(172,313)(173,312)(174,311)(175,310)(176,309)(177,308)(178,307)(179,306)(180,305)(181,304)(182,303)(183,302)(184,301)(185,300)(186,299)(187,298)(188,297)(189,296)(190,295)(191,294)(192,293)(193,292)(194,291)(195,290)(196,289)(197,288)(198,287)(199,286)(200,285)(201,284)(202,283)(203,282)(204,281)(205,280)(206,279)(207,278)(208,277)(209,276)(210,275)(211,274)(212,273)(213,272)(214,271)(215,270)(216,269)(217,268)(218,267)(219,266)(220,265)(221,264)(222,263)(223,262)(224,261)(225,260)(226,259)(227,258)(228,257)(229,256)(230,255)(231,254)(232,253)(233,252)(234,251)(235,250)(236,249)(237,248)(238,247)(239,246)(240,245)(241,244)(242,243)Polycyclic presentation:
2^6,419|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, 385);