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 |
| Heisenberg | H(5,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C7 | C74 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C7 ◁ G | |
| LECS | G ▷ C7 ▷ C1 | |
4453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 343 |
| transitive degree | μt(G) = 343 |
A minimal faithful permutation representation — 4 generators on 343 points:
(1,2,5,13,30,62,102)(3,7,15,32,64,104,150)(4,8,16,33,65,105,149)(6,17,41,82,131,184,148)(9,20,37,68,108,153,200)(10,19,38,69,109,154,201)(11,21,39,70,110,155,199)(12,22,40,71,111,103,151)(14,34,72,121,174,101,18)(23,46,78,116,160,204,243)(24,47,76,117,161,205,244)(25,48,79,118,162,206,242)(26,45,80,119,163,207,245)(27,44,36,73,122,175,227)(28,49,81,120,159,152,202)(29,50,35,74,63,106,156)(31,66,112,164,61,43,42)(51,86,126,170,212,248,278)(52,87,127,167,213,249,279)(53,88,128,171,214,250,277)(54,89,124,172,215,251,280)(55,90,123,169,158,209,253)(56,91,129,168,211,203,246)(57,84,130,173,216,252,281)(58,85,125,114,166,218,262)(59,83,77,67,113,165,217)(60,92,75,115,107,157,208)(93,134,179,222,257,284,306)(94,135,180,223,255,285,307)(95,136,181,224,258,286,305)(96,137,182,219,259,287,308)(97,138,178,220,256,247,282)(98,139,177,225,260,288,309)(99,133,176,221,210,254,290)(100,132,183,226,261,289,310)(140,187,230,265,293,313,326)(141,188,231,266,294,312,327)(142,189,232,267,295,314,325)(143,190,233,268,291,315,328)(144,186,229,264,292,283,311)(145,191,234,263,296,316,329)(146,192,228,269,297,317,330)(147,185,235,270,298,318,331)(193,237,272,300,320,333,338)(194,238,273,301,321,332,339)(195,239,274,302,319,334,340)(196,240,275,299,322,335,341)(197,241,271,303,323,336,342)(198,236,276,304,324,337,343);(2,6,18,43,50,22,8)(5,14,35,16,17,42,40)(7,19,44,83,92,49,21)(13,31,33,34,71,41,74)(15,36,75,39,38,77,81)(20,45,47,85,90,91,48)(30,63,82,111,72,65,66)(32,67,70,73,120,69,115)(37,76,123,79,80,125,129)(46,84,89,87,133,138,88)(62,103,112,131,105,106,121)(64,107,109,159,122,110,113)(68,114,118,117,168,119,169)(78,124,176,128,130,127,178)(86,132,139,137,135,186,136)(102,149,151,156,164,174,184)(104,152,165,154,155,157,175)(108,158,163,211,161,162,166)(116,167,171,172,220,173,221)(126,177,180,181,183,182,229)(134,185,192,191,190,188,189)(150,199,202,208,217,227,201)(153,203,218,207,206,209,205)(160,210,216,256,215,214,213)(170,219,224,225,264,226,223)(179,228,233,232,235,234,231)(187,236,241,240,239,238,237)(200,242,246,253,262,244,245)(204,247,249,252,250,254,251)(212,255,261,292,260,258,259)(222,263,267,269,266,270,268)(230,271,274,272,276,275,273)(243,277,282,290,279,280,281)(248,283,287,289,286,285,288)(257,291,298,294,297,295,296)(265,299,300,303,301,304,302)(278,305,311,307,308,309,310)(284,312,316,318,314,315,317)(293,319,324,321,323,320,322)(306,325,327,328,329,330,331)(313,332,335,337,333,334,336)(326,338,339,340,341,342,343);(1,3,9,23,51,93,140)(2,7,20,46,86,134,187)(4,11,25,53,95,142,193)(5,15,37,78,126,179,230)(6,19,45,84,132,185,236)(8,21,48,88,136,189,237)(10,26,57,100,147,198,148)(12,28,56,97,144,141,194)(13,32,68,116,170,222,265)(14,36,76,124,177,228,271)(16,39,79,128,181,232,272)(17,38,80,130,183,235,276)(18,44,47,89,139,192,241)(22,49,91,138,186,188,238)(24,54,98,146,197,101,27)(29,60,55,99,94,143,195)(30,64,108,160,212,257,293)(31,67,114,167,219,263,299)(33,70,118,171,224,267,300)(34,73,117,172,225,269,303)(35,75,123,176,180,233,274)(40,81,129,178,229,231,273)(41,69,119,173,226,270,304)(42,77,125,127,182,234,275)(43,83,85,87,137,191,240)(50,92,90,133,135,190,239)(52,96,145,196,61,59,58)(62,104,153,204,248,284,313)(63,107,158,210,255,291,319)(65,110,162,214,258,295,320)(66,113,166,213,259,296,322)(71,120,168,220,264,266,301)(72,122,161,215,260,297,323)(74,115,169,221,223,268,302)(82,109,163,216,261,298,324)(102,150,200,243,278,306,326)(103,152,203,247,283,312,332)(105,155,206,250,286,314,333)(106,157,209,254,285,315,334)(111,159,211,256,292,294,321)(112,165,218,249,287,316,335)(121,175,205,251,288,317,336)(131,154,207,252,289,318,337)(149,199,242,277,305,325,338)(151,202,246,282,311,327,339)(156,208,253,290,307,328,340)(164,217,262,279,308,329,341)(174,227,244,280,309,330,342)(184,201,245,281,310,331,343);(3,10,27,59,60,28,11)(7,19,44,83,92,49,21)(9,24,55,25,26,58,56)(15,38,36,77,75,81,39)(20,47,90,48,45,85,91)(23,52,53,54,97,57,99)(32,69,73,67,115,120,70)(37,76,123,79,80,125,129)(46,87,88,89,138,84,133)(51,94,100,144,98,95,96)(64,109,122,113,107,159,110)(68,117,169,118,119,114,168)(78,127,128,124,178,130,176)(86,135,132,186,139,136,137)(93,141,145,147,142,143,146)(104,154,175,165,157,152,155)(108,161,158,162,163,166,211)(116,167,171,172,220,173,221)(126,180,183,229,177,181,182)(134,188,191,185,189,190,192)(140,193,194,195,196,197,198)(150,201,227,217,208,202,199)(153,205,209,206,207,218,203)(160,213,214,215,256,216,210)(170,223,226,264,225,224,219)(179,231,234,235,232,233,228)(187,237,238,239,240,241,236)(200,244,253,242,245,262,246)(204,249,250,251,247,252,254)(212,255,261,292,260,258,259)(222,266,263,270,267,268,269)(230,272,273,274,275,271,276)(243,279,277,280,282,281,290)(248,285,289,283,288,286,287)(257,294,296,298,295,291,297)(265,300,301,302,299,303,304)(278,307,310,311,309,305,308)(284,312,316,318,314,315,317)(293,320,321,319,322,323,324)(306,327,329,331,325,328,330)(313,333,332,334,335,336,337)(326,338,339,340,341,342,343)Polycyclic presentation:
7^5|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, 78);