Finite Group DatabasebetaFinite Group Databasebeta| order | 16807 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C73 · Schur multiplier (H2) = C7 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C72 | C73 |
| prime radical | C73 | C72 |
| LCS | G ▷ C72 ▷ C1 | |
| UCS | C1 ◁ C72 ◁ G | |
| LECS | G ▷ C72 ▷ C1 | |
203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 392 |
| transitive degree | μt(G) = 2401 |
A minimal faithful permutation representation — 3 generators on 392 points:
(2,4,7,12,20,32,42)(5,9,15,24,8,13,21)(10,17,27,16,25,14,22)(18,29,23,33,26,35,28)(30,39,37,34,43,38,36)(40,49,48,47,46,45,44)(50,51,54,62,80,115,164)(52,58,72,101,146,201,259)(53,57,65,83,118,167,217)(55,66,90,133,187,248,216)(56,69,96,139,190,250,272)(59,75,106,150,175,230,283)(60,74,102,120,173,228,218)(61,71,89,123,171,165,219)(63,84,124,178,236,163,67)(64,87,129,180,237,215,68)(70,98,142,194,224,277,271)(73,103,147,170,225,279,312)(76,109,154,184,242,278,318)(77,108,128,122,176,232,286)(78,105,134,125,179,166,220)(79,100,85,126,116,168,222)(81,119,172,227,114,93,91)(82,121,174,229,162,94,92)(86,127,117,169,223,113,136)(88,131,183,241,274,161,137)(95,97,140,191,221,275,214)(99,144,197,233,287,314,270)(104,148,177,234,289,315,311)(107,151,182,239,276,316,344)(110,157,193,251,288,328,348)(111,156,132,185,244,280,320)(112,153,135,130,181,238,273)(138,143,195,231,284,313,213)(141,192,226,281,322,160,188)(145,199,240,293,321,353,269)(149,202,243,295,323,345,310)(152,204,245,282,324,346,343)(155,186,246,285,326,347,368)(158,209,255,294,334,356,371)(159,208,196,253,290,317,349)(189,198,235,291,319,351,212)(200,257,297,327,359,373,268)(203,260,298,329,350,374,309)(205,247,299,330,352,376,342)(206,252,301,331,354,369,367)(207,254,292,332,355,370,384)(210,266,306,338,362,378,386)(211,249,256,296,325,357,372)(258,307,339,363,379,387,267)(261,300,340,364,380,388,308)(262,302,333,365,381,389,341)(263,303,335,358,382,390,366)(264,304,336,360,375,391,383)(265,305,337,361,377,385,392);(1,2,5,10,18,30,40,3,4,8,14,23,34,44,6,7,9,16,26,36,45,11,12,13,17,28,37,46,19,20,15,22,29,38,47,31,32,21,25,33,39,48,41,42,24,27,35,43,49)(51,55,67,93,100,71,57)(54,63,85,65,66,91,89)(56,68,94,136,105,74,58)(62,81,83,84,123,90,126)(64,86,102,69,92,134,72)(70,95,137,153,108,75,73)(80,116,133,171,124,118,119)(82,120,87,125,96,127,101)(88,128,103,97,135,106,98)(99,138,188,156,109,107,104)(115,165,172,187,167,168,178)(117,139,179,129,173,121,146)(122,142,130,147,131,150,140)(132,151,144,141,154,148,143)(145,189,208,157,155,152,149)(164,217,219,222,227,236,248)(166,174,190,228,169,180,201)(170,181,194,176,191,175,183)(177,192,182,195,184,197,185)(186,202,198,193,204,199,196)(200,249,209,207,206,205,203)(218,220,223,229,237,250,259)(221,225,232,241,224,230,238)(226,234,244,233,242,231,239)(235,246,240,251,243,253,245)(247,257,255,252,260,256,254)(258,266,265,264,263,262,261)(273,274,275,277,279,283,286)(276,280,284,289,278,281,287)(282,290,295,288,293,285,291)(292,298,294,299,296,301,297)(300,306,304,302,307,305,303)(313,314,315,316,318,320,322)(317,321,324,328,319,323,326)(325,330,334,329,332,327,331)(333,338,335,339,336,340,337)(345,346,347,348,349,351,353)(350,354,356,359,352,355,357)(358,362,365,361,364,360,363)(369,370,371,372,373,374,376)(375,378,380,382,377,379,381)(385,386,387,388,389,390,391);(50,52,59,76,110,158,210,53,60,77,111,159,211,267,61,78,112,160,212,268,308,79,113,161,213,269,309,341,114,162,214,270,310,342,366,163,215,271,311,343,367,383,216,272,312,344,368,384,392)(51,56,70,99,145,200,258,57,58,73,104,149,203,261,71,74,75,107,152,205,262,100,105,108,109,155,206,263,93,136,153,156,157,207,264,67,94,137,188,208,209,265,55,68,95,138,189,249,266)(54,64,88,132,186,247,300,65,69,97,141,193,252,302,89,72,98,143,196,254,303,85,102,103,144,198,255,304,91,134,106,148,199,256,305,63,86,128,151,202,257,306,66,92,135,154,204,260,307)(62,82,122,177,235,292,333,83,87,130,182,240,294,335,123,96,131,184,243,296,336,126,101,140,185,245,297,337,81,120,142,192,246,298,338,84,125,147,195,251,299,339,90,127,150,197,253,301,340)(80,117,170,226,282,325,358,118,121,175,231,285,327,360,171,129,176,233,288,329,361,116,139,181,234,290,330,362,119,146,183,239,291,331,363,124,173,191,242,293,332,364,133,179,194,244,295,334,365)(115,166,221,276,317,350,375,167,169,224,278,319,352,377,165,174,225,280,321,354,378,168,180,230,281,323,355,379,172,190,232,284,324,356,380,178,201,238,287,326,357,381,187,228,241,289,328,359,382)(164,218,273,313,345,369,385,217,220,274,314,346,370,386,219,223,275,315,347,371,387,222,229,277,316,348,372,388,227,237,279,318,349,373,389,236,250,283,320,351,374,390,248,259,286,322,353,376,391)Polycyclic presentation:
7^5|b^7=d,c^7=e,b^a=bde,c^a=ce679 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16807, 55);