Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C73 · Schur multiplier (H2) = C73 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C49 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C73 |
| socle | C7 | C7×H(3,𝔽7) |
| prime radical | H(3,𝔽7)⋊C7 | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C49 ◁ C7×C49 ◁ G | |
| LECS | G ▷ C72 ▷ C7 ▷ C1 | |
259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 343 |
| transitive degree | μt(G) = 343 |
A minimal faithful permutation representation — 3 generators on 343 points:
(2,5,12,27,46,22,8)(6,15,19,18,35,13,29)(7,14,28,52,79,45,21)(16,36,64,54,33,42,30)(17,34,41,40,63,31,55)(20,32,53,86,121,78,44)(37,60,65,75,92,56,69)(38,66,101,88,61,74,57)(39,62,73,72,100,58,89)(43,59,87,129,170,120,77)(67,90,117,96,110,107,102)(68,97,103,116,136,91,109)(70,104,146,131,98,115,93)(71,99,114,113,145,94,132)(76,95,130,177,217,169,119)(106,134,165,141,157,153,148)(108,142,149,164,184,135,156)(111,150,194,179,143,163,137)(112,144,162,161,193,138,180)(118,139,178,223,260,216,168)(152,182,213,189,205,201,196)(155,190,197,212,230,183,204)(159,198,239,225,191,211,185)(160,192,210,209,238,186,226)(167,187,224,265,295,259,215)(200,228,257,234,250,246,241)(203,235,242,256,271,229,249)(207,243,278,266,236,255,231)(208,237,254,253,277,232,267)(245,269,293,274,289,285,280)(248,275,281,292,303,270,288)(252,282,307,299,276,291,272)(284,301,319,305,316,313,309)(287,306,310,318,325,302,315)(312,324,335,327,333,331,329);(1,2,6,16,37,67,105)(3,7,17,38,68,106,151)(4,8,19,36,65,102,147)(5,13,30,56,90,133,128)(9,20,39,70,108,152,199)(10,21,41,66,103,148,195)(11,22,35,64,92,107,154)(12,18,42,75,117,166,85)(14,31,57,91,134,181,176)(15,33,60,96,140,51,27)(23,43,71,111,155,200,244)(24,44,73,104,149,196,240)(25,45,63,101,136,153,202)(26,46,29,54,69,110,158)(28,40,74,116,165,214,127)(32,58,93,135,182,227,222)(34,61,97,141,188,84,52)(47,76,112,159,203,245,283)(48,77,114,150,197,241,279)(49,78,100,146,184,201,247)(50,79,55,88,109,157,206)(53,72,115,164,213,258,175)(59,94,137,183,228,268,264)(62,98,142,189,233,126,86)(80,118,160,207,248,284,311)(81,119,162,198,242,280,308)(82,120,145,194,230,246,286)(83,121,89,131,156,205,251)(87,113,163,212,257,294,221)(95,138,185,229,269,300,298)(99,143,190,234,273,174,129)(122,167,208,252,287,312,330)(123,168,210,243,281,309,328)(124,169,193,239,271,285,314)(125,170,132,179,204,250,290)(130,161,211,256,293,320,263)(139,186,231,270,301,323,322)(144,191,235,274,304,220,177)(171,215,254,282,310,329,340)(172,216,238,278,303,313,332)(173,217,180,225,249,289,317)(178,209,255,292,319,336,297)(187,232,272,302,324,338,337)(192,236,275,305,326,262,223)(218,259,277,307,325,331,341)(219,260,226,266,288,316,334)(224,253,291,318,335,343,321)(237,276,306,327,339,296,265)(261,295,267,299,315,333,342);(1,3,9,23,47,80,122,4,10,24,48,81,123,171,11,25,49,82,124,172,218,26,50,83,125,173,219,261,51,84,126,174,220,262,296,85,127,175,221,263,297,321,128,176,222,264,298,322,337)(2,7,20,43,76,118,167,8,21,44,77,119,168,215,22,45,78,120,169,216,259,46,79,121,170,217,260,295,27,52,86,129,177,223,265,12,28,53,87,130,178,224,5,14,32,59,95,139,187)(6,17,39,71,112,160,208,19,41,73,114,162,210,254,35,63,100,145,193,238,277,29,55,89,132,180,226,267,15,34,62,99,144,192,237,18,40,72,113,161,209,253,13,31,58,94,138,186,232)(16,38,70,111,159,207,252,36,66,104,150,198,243,282,64,101,146,194,239,278,307,54,88,131,179,225,266,299,33,61,98,143,191,236,276,42,74,115,163,211,255,291,30,57,93,137,185,231,272)(37,68,108,155,203,248,287,65,103,149,197,242,281,310,92,136,184,230,271,303,325,69,109,156,204,249,288,315,60,97,142,190,235,275,306,75,116,164,212,256,292,318,56,91,135,183,229,270,302)(67,106,152,200,245,284,312,102,148,196,241,280,309,329,107,153,201,246,285,313,331,110,157,205,250,289,316,333,96,141,189,234,274,305,327,117,165,213,257,293,319,335,90,134,182,228,269,301,324)(105,151,199,244,283,311,330,147,195,240,279,308,328,340,154,202,247,286,314,332,341,158,206,251,290,317,334,342,140,188,233,273,304,326,339,166,214,258,294,320,336,343,133,181,227,268,300,323,338)Polycyclic presentation:
7^5|c^7=e,b^a=bd,d^b=de679 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 68);