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:
(1,2,5,14,35,79,128)(3,9,24,57,109,164,223)(4,8,17,38,82,131,182)(6,18,45,96,152,211,12)(7,21,53,108,163,222,180)(10,27,63,97,153,188,241)(11,26,47,90,139,185,181)(13,23,44,87,136,130,186)(15,39,88,143,201,33,20)(16,42,94,151,210,127,22)(19,48,101,144,202,237,72)(25,60,84,140,196,240,73)(28,67,117,155,197,248,275)(29,66,46,99,133,189,243)(30,50,103,89,145,184,238)(31,65,40,91,147,193,247)(32,62,59,98,134,129,183)(34,56,43,95,81,135,191)(36,83,137,192,77,52,41)(37,86,142,200,78,55,54)(49,104,158,204,244,279,74)(51,58,110,138,194,236,75)(61,92,149,207,249,276,71)(64,111,146,190,245,283,125)(68,115,159,212,259,281,308)(69,106,100,141,198,252,280)(70,116,112,148,203,242,274)(76,107,93,85,80,132,187)(102,154,195,239,277,123,114)(105,157,208,255,290,303,120)(113,166,199,251,284,307,176)(118,165,216,250,287,302,122)(119,167,224,258,294,318,326)(121,162,170,205,256,278,304)(124,168,156,213,246,282,305)(126,161,150,206,253,286,311)(160,214,262,289,315,332,174)(169,225,266,291,306,324,178)(171,209,257,293,310,329,173)(172,218,261,292,313,327,337)(175,219,217,254,288,314,325)(177,220,228,260,285,309,328)(179,221,215,263,297,312,323)(226,264,295,316,330,338,235)(227,265,296,317,331,339,233)(229,267,298,319,333,340,230)(231,268,273,301,322,336,343)(232,269,272,300,321,335,342)(234,270,271,299,320,334,341);(1,3,10,28,68,119,172,4,12,30,70,121,174,230,13,33,73,123,176,173,231,34,77,72,125,120,175,232,78,76,75,69,122,177,233,127,32,29,71,124,178,234,180,11,31,74,126,179,235)(2,6,19,49,105,160,218,8,20,51,67,118,171,229,23,52,66,116,168,219,268,56,107,65,114,161,220,269,55,62,27,64,115,169,227,22,26,50,106,162,221,270,7,9,25,61,113,167,226)(5,15,40,92,150,209,261,17,41,63,104,159,217,267,44,93,103,117,170,228,273,43,59,60,112,166,225,272,54,47,48,102,157,215,265,16,24,58,111,165,224,271,21,18,46,100,156,214,264)(14,36,84,141,199,254,292,38,85,101,149,208,260,298,87,98,110,158,216,266,301,95,90,99,155,213,263,300,37,57,91,148,206,258,296,42,45,97,154,212,262,299,53,39,89,146,205,257,295)(35,80,133,190,246,285,313,82,134,147,198,253,291,319,136,139,153,207,259,297,322,81,109,145,204,256,294,321,86,96,140,197,251,289,317,94,88,144,203,255,293,320,108,83,138,195,250,288,316)(79,129,184,239,278,306,327,131,185,196,245,284,312,333,130,164,202,252,290,318,336,135,152,194,249,287,315,335,142,143,189,244,282,310,331,151,137,193,248,286,314,334,163,132,188,242,281,309,330)(128,181,236,274,302,323,337,182,223,243,277,305,326,340,186,211,247,283,311,332,343,191,201,241,280,308,329,342,200,192,238,276,304,325,339,210,187,240,279,307,328,341,222,183,237,275,303,324,338);(2,7,22,55,56,23,8)(5,16,43,17,21,54,44)(6,9,26,62,107,52,20)(14,37,38,42,87,53,95)(15,24,59,41,18,47,93)(19,25,50,27,65,66,51)(35,81,108,136,94,82,86)(36,57,85,45,98,39,90)(40,58,60,63,46,48,103)(49,61,106,64,114,116,67)(79,130,142,163,131,135,151)(80,109,83,139,88,134,96)(84,91,101,97,110,89,99)(92,111,112,104,100,102,117)(105,113,162,115,161,168,118)(128,182,186,191,200,210,222)(129,164,143,132,185,152,137)(133,145,138,153,144,147,140)(141,148,149,154,158,146,155)(150,165,166,159,156,157,170)(160,167,221,169,220,219,171)(181,223,211,201,192,187,183)(184,202,189,188,196,194,193)(190,204,195,207,203,198,197)(199,206,208,212,216,205,213)(209,224,225,217,214,215,228)(218,226,270,227,269,268,229)(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|b^7=e,b^a=bd,c^a=ce,d^b=de391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 74);