Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C72 · Schur multiplier (H2) = C7 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C72 | H(3,𝔽7) |
| commutator, Frattini, cosocle | C73 | C72 |
| prime radical | C7×H(3,𝔽7) | C7 |
| LCS | G ▷ C73 ▷ C72 ▷ C1 | |
| UCS | C1 ◁ C72 ◁ C73 ◁ G | |
| LECS | G ▷ C73 ▷ C72 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 392 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 2 generators on 392 points:
(2,4,8,16,24,14,6)(5,10,13,12,20,9,17)(11,21,29,26,19,23,18)(22,28,30,33,36,27,32)(31,35,42,37,41,40,38)(50,51,54,62,80,117,163)(52,58,72,102,145,197,215)(53,57,65,83,120,166,216)(55,66,90,131,164,217,60)(56,69,98,144,196,254,214)(59,75,108,132,184,229,270)(61,71,89,124,171,165,220)(63,84,118,167,221,115,97)(64,87,129,182,241,162,70)(67,94,125,173,218,273,114)(68,93,134,174,219,272,78)(73,105,148,200,255,271,77)(74,92,81,121,172,228,269)(76,111,136,188,242,281,313)(79,101,88,130,119,170,227)(82,123,177,235,116,100,99)(85,127,168,225,280,312,110)(86,126,169,224,276,161,143)(91,135,183,222,277,212,142)(95,139,187,243,274,316,211)(96,103,146,198,256,275,160)(104,133,186,223,279,213,107)(106,128,180,226,282,315,159)(109,151,178,232,284,314,113)(112,156,201,236,290,328,345)(122,175,231,286,268,153,137)(138,179,230,278,319,311,195)(140,192,237,288,317,349,310)(141,149,176,233,289,318,267)(147,185,244,287,321,344,155)(150,189,234,291,323,348,266)(152,199,245,300,326,347,210)(154,190,238,285,322,346,158)(157,207,250,295,329,353,369)(181,239,283,324,351,343,194)(191,246,299,320,352,368,206)(193,251,301,340,350,374,367)(202,248,296,327,355,373,342)(203,257,294,330,359,372,309)(204,249,292,332,354,371,265)(205,240,297,325,357,370,209)(208,263,303,338,360,380,385)(247,293,331,356,376,384,253)(252,307,336,362,375,391,392)(258,304,334,364,379,390,383)(259,298,339,358,382,389,366)(260,305,337,361,378,388,341)(261,302,335,363,381,387,308)(262,306,333,365,377,386,264);(1,2,5,11,22,31,39)(3,6,13,21,30,38,46)(4,9,18,27,35,44,43)(7,14,20,29,36,40,47)(8,12,23,33,42,49,34)(10,19,28,37,45,25,16)(15,24,17,26,32,41,48)(50,52,59,76,112,157,208,79,115,160,211,266,309,341,214,269,312,344,368,384,392,61,78,114,159,210,265,308,162,213,268,311,343,367,383,53,60,77,113,158,209,264,116,161,212,267,310,342,366)(51,55,67,95,140,193,252,101,143,153,155,156,205,261,56,58,73,106,150,202,258,71,97,142,195,206,207,262,70,74,75,109,152,203,259,57,68,96,141,194,253,263,100,107,110,111,154,204,260)(54,63,85,128,181,240,298,88,92,94,138,190,248,303,69,93,137,151,192,250,305,89,104,105,149,201,257,307,64,66,91,136,189,247,302,65,86,108,139,191,249,304,99,72,103,147,199,251,306)(62,81,122,176,234,292,333,130,134,148,188,246,301,339,98,133,135,187,245,297,338,124,90,132,185,239,296,337,87,126,146,180,238,295,336,83,102,127,179,237,294,335,82,84,125,178,236,293,334)(80,118,168,226,283,325,358,119,121,173,230,285,327,360,144,174,175,232,288,329,361,171,186,200,233,290,330,362,129,131,183,242,291,331,363,120,169,184,243,299,332,364,123,145,198,244,300,340,365)(117,164,218,274,317,350,375,170,224,231,287,328,357,381,196,197,255,282,323,355,379,165,167,222,278,320,353,377,182,172,229,284,326,359,382,166,219,256,289,324,356,380,177,223,225,281,322,354,378)(163,215,270,313,345,369,385,227,221,275,316,348,372,388,254,228,280,321,352,376,391,220,272,273,315,347,371,387,241,279,286,319,351,374,390,216,217,271,314,346,370,386,235,276,277,318,349,373,389)Polycyclic presentation:
7^5|b^7=d^3,b^a=bc,c^a=cd,c^b=ce391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 6);