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, socle | C7 | C7×H(3,𝔽7) |
| commutator, Frattini, cosocle | C72 | C73 |
| prime radical | C7×H(3,𝔽7) | C7 |
| LCS | G ▷ C72 ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7 ◁ C73 ◁ G | |
| LECS | G ▷ C72 ▷ C7 ▷ C1 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 343 |
| transitive degree | μt(G) = 343 |
A minimal faithful permutation representation — 3 generators on 343 points:
(1,2,5,14,33,71,126,4,8,16,36,74,129,181,13,22,39,80,73,132,190,32,50,86,78,133,191,180,70,7,20,47,100,161,221,125,21,48,35,77,139,200,178,49,101,79,140,128,184)(3,9,23,51,102,162,222,12,25,53,89,143,127,182,31,56,106,34,75,134,192,69,6,17,40,87,150,210,11,19,42,83,136,187,179,30,46,92,103,72,130,185,68,99,15,37,81,141,201)(10,26,57,88,151,186,241,28,60,41,90,131,188,243,65,18,43,93,142,202,237,122,45,95,82,144,183,238,29,59,38,84,146,193,247,67,112,52,104,135,194,236,124,24,54,76,137,196,240)(27,61,113,153,197,248,275,64,55,85,148,207,249,276,121,58,105,145,189,245,283,119,111,107,147,203,242,274,66,44,96,156,204,244,279,63,98,91,138,198,252,280,120,109,94,152,195,239,277)(62,110,157,211,259,281,308,117,166,154,212,246,282,305,116,97,155,208,255,290,303,174,160,168,205,256,278,304,123,159,149,206,253,286,311,118,114,163,215,250,287,302,176,108,164,199,251,284,307)(115,165,223,258,294,318,326,172,167,224,266,291,306,324,173,218,216,254,288,314,325,233,158,213,262,289,315,332,177,219,214,263,297,312,323,175,220,227,260,285,309,328,171,169,209,257,293,310,329)(170,217,261,292,313,327,337,230,268,271,299,320,334,341,232,270,272,300,321,335,342,229,228,267,298,319,333,340,235,225,264,295,316,330,338,234,226,265,296,317,331,339,231,269,273,301,322,336,343);(1,3,10,27,62,115,170)(2,6,18,44,97,158,217)(4,12,28,64,117,172,230)(5,15,38,85,149,209,261)(7,9,24,55,108,165,225)(8,19,45,98,160,219,268)(11,29,66,123,177,235,70)(13,31,65,121,116,173,232)(14,34,76,138,199,254,292)(16,23,52,105,163,223,271)(17,41,91,154,213,264,20)(21,25,26,58,110,167,226)(22,46,59,109,159,220,270)(30,67,63,118,175,234,125)(32,69,122,119,174,233,229)(33,72,131,189,246,285,313)(35,51,84,147,206,258,296)(36,40,88,152,211,262,299)(37,82,145,205,257,295,47)(39,53,54,107,164,224,272)(42,43,94,155,214,265,48)(49,56,60,111,166,218,269)(50,99,112,61,114,169,228)(57,96,157,216,267,86,106)(68,124,120,176,171,231,178)(71,127,183,239,278,306,327)(73,102,144,204,256,294,321)(74,81,142,203,255,293,320)(75,135,195,250,288,316,100)(77,87,137,197,251,289,317)(78,103,93,148,208,260,298)(79,89,104,156,215,266,301)(80,83,90,153,212,263,300)(92,95,113,168,227,273,101)(126,179,236,274,302,323,337)(128,162,202,252,290,318,336)(129,134,193,248,286,314,334)(130,186,242,281,309,330,161)(132,150,194,249,287,315,335)(133,143,146,198,253,291,319)(136,151,207,259,297,322,140)(139,141,188,244,282,310,331)(180,222,243,277,305,326,340)(181,185,240,279,307,328,341)(182,237,275,303,324,338,221)(184,210,247,283,311,332,343)(187,196,245,284,312,333,191)(190,201,241,280,308,329,342)(192,238,276,304,325,339,200);(2,7,8,21,22,49,50)(5,16,39,86,20,48,101)(6,9,19,25,46,56,99)(14,35,78,36,79,47,80)(15,23,53,106,17,42,92)(18,24,45,26,59,60,112)(33,73,100,140,74,133,77)(34,51,103,40,89,37,83)(38,52,54,57,41,43,95)(44,55,98,58,109,111,61)(71,128,139,161,191,132,129)(72,102,75,136,81,143,87)(76,84,93,88,104,82,90)(85,105,107,96,91,94,113)(97,108,160,110,159,166,114)(126,180,184,190,200,181,221)(127,162,141,130,187,150,134)(131,144,135,151,142,146,137)(138,147,148,152,156,145,153)(149,163,164,157,154,155,168)(158,165,219,167,220,218,169)(179,222,210,201,192,185,182)(183,202,188,186,196,194,193)(189,204,195,207,203,198,197)(199,206,208,211,215,205,212)(209,223,224,216,213,214,227)(217,225,268,226,270,269,228)(236,243,247,241,238,240,237)(239,252,244,242,245,249,248)(246,256,250,259,255,253,251)(254,258,260,262,266,257,263)(261,271,272,267,264,265,273)(274,277,283,280,276,279,275)(278,290,282,281,284,287,286)(285,294,288,297,293,291,289)(292,296,298,299,301,295,300)(302,305,311,308,304,307,303)(306,318,310,309,312,315,314)(313,321,316,322,320,319,317)(323,326,332,329,325,328,324)(327,336,331,330,333,335,334)(337,340,343,342,339,341,338)Polycyclic presentation:
7^5|a^7=e,b^a=bd,c^a=ce^3,d^b=de^3391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 76);