Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C74 · Schur multiplier (H2) = C75 |
| extraspecial | 7−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C7 | C74 |
| prime radical | C7×H(3,𝔽7) | C7 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
3718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 343 |
| transitive degree | μt(G) = 343 |
A minimal faithful permutation representation — 4 generators on 343 points:
(1,2,5,12,27,56,100)(3,7,14,29,58,102,150)(4,8,15,30,59,103,149)(6,16,37,74,127,182,148)(9,19,34,62,106,153,203)(10,20,35,63,107,154,204)(11,21,36,64,108,101,151)(13,31,65,116,171,99,17)(18,39,75,128,183,233,195)(22,43,70,112,158,152,205)(23,44,71,113,159,207,244)(24,45,72,114,160,206,245)(25,46,73,115,161,208,246)(26,47,32,67,57,104,155)(28,60,109,162,55,40,38)(33,66,117,172,223,139,41)(42,77,129,184,234,274,92)(48,81,68,119,105,156,209)(49,82,120,166,157,210,251)(50,83,122,167,212,248,278)(51,84,123,111,164,216,257)(52,85,124,168,213,247,279)(53,86,125,169,211,249,280)(54,87,126,170,214,250,281)(61,110,163,215,88,78,76)(69,118,173,224,265,89,79)(80,130,185,235,275,302,95)(90,133,176,165,217,258,293)(91,134,177,219,253,283,305)(93,131,121,174,225,266,300)(94,135,178,220,254,282,306)(96,136,179,221,252,284,307)(97,137,180,218,255,285,308)(98,138,181,222,256,286,309)(132,186,236,276,303,323,142)(140,187,175,226,267,301,322)(141,189,228,260,288,311,325)(143,190,229,261,289,310,326)(144,191,230,262,287,312,327)(145,192,231,259,290,313,328)(146,193,227,263,291,314,329)(147,194,232,264,292,315,330)(188,237,277,304,324,337,196)(197,239,269,295,317,332,338)(198,240,270,296,318,331,339)(199,241,271,297,316,333,340)(200,242,272,294,319,334,341)(201,243,268,298,320,335,342)(202,238,273,299,321,336,343);(2,6,17,40,47,21,8)(5,13,32,15,16,38,36)(7,18,41,78,81,43,19)(12,28,30,31,64,37,67)(14,33,68,34,39,76,70)(20,42,79,84,82,45,44)(27,57,74,108,65,59,60)(29,61,62,66,112,75,119)(35,69,120,71,77,123,72)(46,80,131,133,86,85,83)(56,101,109,127,103,104,116)(58,105,128,158,117,106,110)(63,111,113,118,114,129,166)(73,121,125,122,130,176,124)(87,132,187,137,136,135,134)(100,149,151,155,162,171,182)(102,152,163,183,153,156,172)(107,157,184,160,173,159,164)(115,165,167,174,168,185,169)(126,175,179,177,186,180,178)(138,188,193,192,191,190,189)(150,203,205,209,215,223,233)(154,206,216,234,207,210,224)(161,211,235,213,225,212,217)(170,218,219,226,220,236,221)(181,227,230,228,237,231,229)(194,238,243,242,241,240,239)(204,244,245,251,257,265,274)(208,247,258,275,248,249,266)(214,252,276,254,267,253,255)(222,259,260,263,261,277,262)(232,268,271,269,273,272,270)(246,278,279,280,293,300,302)(250,282,285,303,283,284,301)(256,287,304,289,291,288,290)(264,294,295,298,296,299,297)(281,305,306,307,308,322,323)(286,310,313,324,311,312,314)(292,316,321,318,320,317,319)(309,325,326,327,328,329,337)(315,331,334,336,332,333,335)(330,338,339,340,341,342,343);(3,9,22,48,88,139,195)(7,19,43,81,78,41,18)(10,24,51,92,23,49,89)(14,34,70,68,76,33,39)(20,45,84,42,44,82,79)(25,53,95,52,93,50,90)(29,62,112,119,61,66,75)(35,72,123,77,71,120,69)(46,86,80,85,131,83,133)(54,97,91,140,94,142,96)(58,106,158,105,110,117,128)(63,114,111,129,113,166,118)(73,125,130,124,121,122,176)(87,137,134,187,135,132,136)(98,146,144,141,196,145,143)(102,153,152,156,163,172,183)(107,160,164,184,159,157,173)(115,169,185,168,174,167,165)(126,180,177,175,178,186,179)(138,193,191,189,188,192,190)(147,202,201,200,199,198,197)(150,203,205,209,215,223,233)(154,206,216,234,207,210,224)(161,211,235,213,225,212,217)(170,218,219,226,220,236,221)(181,227,230,228,237,231,229)(194,238,243,242,241,240,239)(204,245,257,274,244,251,265)(208,249,275,247,266,248,258)(214,255,253,267,254,276,252)(222,263,262,260,277,259,261)(232,273,268,272,271,270,269)(246,280,302,279,300,278,293)(250,285,283,301,282,303,284)(256,291,287,288,304,290,289)(264,299,298,294,297,296,295)(281,308,305,322,306,323,307)(286,314,312,311,324,313,310)(292,321,320,319,316,318,317)(309,329,327,325,337,328,326)(315,336,335,334,333,331,332)(330,343,342,341,340,339,338);(1,3,10,25,54,98,147,4,9,23,50,91,141,197,11,22,24,52,94,143,198,26,48,49,53,96,144,199,55,88,51,90,97,145,200,99,139,89,93,140,146,201,148,195,92,95,142,196,202)(2,7,20,46,87,138,194,8,19,44,83,134,189,239,21,43,45,85,135,190,240,47,81,82,86,136,191,241,40,78,84,133,137,192,242,17,41,79,131,187,193,243,6,18,42,80,132,188,238)(5,14,35,73,126,181,232,15,34,71,122,177,228,269,36,70,72,124,178,229,270,32,68,120,125,179,230,271,38,76,123,176,180,231,272,13,33,69,121,175,227,268,16,39,77,130,186,237,273)(12,29,63,115,170,222,264,30,62,113,167,219,260,295,64,112,114,168,220,261,296,67,119,166,169,221,262,297,28,61,111,165,218,259,294,31,66,118,174,226,263,298,37,75,129,185,236,277,299)(27,58,107,161,214,256,292,59,106,159,212,253,288,317,108,158,160,213,254,289,318,57,105,157,211,252,287,316,60,110,164,217,255,290,319,65,117,173,225,267,291,320,74,128,184,235,276,304,321)(56,102,154,208,250,286,315,103,153,207,248,283,311,332,101,152,206,247,282,310,331,104,156,210,249,284,312,333,109,163,216,258,285,313,334,116,172,224,266,301,314,335,127,183,234,275,303,324,336)(100,150,204,246,281,309,330,149,203,244,278,305,325,338,151,205,245,279,306,326,339,155,209,251,280,307,327,340,162,215,257,293,308,328,341,171,223,265,300,322,329,342,182,233,274,302,323,337,343)Polycyclic presentation:
7^5|d^7=e,b^a=be,d^c=de2407 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 82);