Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C73 · Schur multiplier (H2) = C73 |
| direct product | C7×#14558879 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C72 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C73 |
| prime radical | C7×H(3,𝔽7) | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C72 ◁ C73 ◁ G | |
| LECS | G ▷ C72 ▷ C7 ▷ C1 | |
272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 350 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 3 generators on 350 points:
(1,2,3,4,5,6,7);(8,9,12,20,38,75,121)(10,16,30,60,103,155,173)(11,15,23,41,78,124,174)(13,24,48,89,122,175,18)(14,27,56,102,154,212,172)(17,33,66,90,142,187,228)(19,29,47,82,129,123,178)(21,42,76,125,179,73,55)(22,45,87,140,199,120,28)(25,52,83,131,176,231,72)(26,51,92,132,177,230,36)(31,63,106,158,213,229,35)(32,50,39,79,130,186,227)(34,69,94,146,200,239,271)(37,59,46,88,77,128,185)(40,81,135,193,74,58,57)(43,85,126,183,238,270,68)(44,84,127,182,234,119,101)(49,93,141,180,235,170,100)(53,97,145,201,232,274,169)(54,61,104,156,214,233,118)(62,91,144,181,237,171,65)(64,86,138,184,240,273,117)(67,109,136,190,242,272,71)(70,114,159,194,248,286,303)(80,133,189,244,226,111,95)(96,137,188,236,277,269,153)(98,150,195,246,275,307,268)(99,107,134,191,247,276,225)(105,143,202,245,279,302,113)(108,147,192,249,281,306,224)(110,157,203,258,284,305,168)(112,148,196,243,280,304,116)(115,165,208,253,287,311,327)(139,197,241,282,309,301,152)(149,204,257,278,310,326,164)(151,209,259,298,308,332,325)(160,206,254,285,313,331,300)(161,215,252,288,317,330,267)(162,207,250,290,312,329,223)(163,198,255,283,315,328,167)(166,221,261,296,318,338,343)(205,251,289,314,334,342,211)(210,265,294,320,333,349,350)(216,262,292,322,337,348,341)(217,256,297,316,340,347,324)(218,263,295,319,336,346,299)(219,260,293,321,339,345,266)(220,264,291,323,335,344,222);(8,10,17,34,70,115,166,37,73,118,169,224,267,299,172,227,270,302,326,342,350,19,36,72,117,168,223,266,120,171,226,269,301,325,341,11,18,35,71,116,167,222,74,119,170,225,268,300,324)(9,13,25,53,98,151,210,59,101,111,113,114,163,219,14,16,31,64,108,160,216,29,55,100,153,164,165,220,28,32,33,67,110,161,217,15,26,54,99,152,211,221,58,65,68,69,112,162,218)(12,21,43,86,139,198,256,46,50,52,96,148,206,261,27,51,95,109,150,208,263,47,62,63,107,159,215,265,22,24,49,94,147,205,260,23,44,66,97,149,207,262,57,30,61,105,157,209,264)(20,39,80,134,192,250,291,88,92,106,146,204,259,297,56,91,93,145,203,255,296,82,48,90,143,197,254,295,45,84,104,138,196,253,294,41,60,85,137,195,252,293,40,42,83,136,194,251,292)(38,76,126,184,241,283,316,77,79,131,188,243,285,318,102,132,133,190,246,287,319,129,144,158,191,248,288,320,87,89,141,200,249,289,321,78,127,142,201,257,290,322,81,103,156,202,258,298,323)(75,122,176,232,275,308,333,128,182,189,245,286,315,339,154,155,213,240,281,313,337,123,125,180,236,278,311,335,140,130,187,242,284,317,340,124,177,214,247,282,314,338,135,181,183,239,280,312,336)(121,173,228,271,303,327,343,185,179,233,274,306,330,346,212,186,238,279,310,334,349,178,230,231,273,305,329,345,199,237,244,277,309,332,348,174,175,229,272,304,328,344,193,234,235,276,307,331,347)679 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 70);