Finite Group DatabasebetaFinite Group Databasebeta| order | 34902 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26042767 |
| Fitting, socle, derived | C3×C7×C277 ⋊ C2×C3 |
| Hall | C3×#19154113 ⋊ C2 |
| Hall | C7×C277 ⋊ C3×D3 |
| Sylow | C277 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ C3⋊(D277⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C277 | C2×C3 |
| LCS | G ▷ C3×C7×C277 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 287 |
| transitive degree | μt(G) = 5817 |
A minimal faithful permutation representation — 2 generators on 287 points:
(2,3)(5,8,9,6,7,10)(12,16,17,13,14,18)(15,24,25,19,20,34)(21,37,38,27,28,51)(22,36,32,30,26,23)(29,56,57,40,41,74)(31,54,55,43,39,48)(33,35,53,45,46,49)(42,79,80,59,60,103)(44,77,78,62,58,71)(47,52,76,64,65,72)(50,92,90,68,69,67)(61,108,109,82,83,138)(63,106,107,85,81,100)(66,75,105,87,88,101)(70,121,122,93,94,96)(73,127,119,97,98,91)(84,143,144,111,112,179)(86,141,142,114,110,135)(89,104,140,116,117,136)(95,156,157,123,124,131)(99,163,161,128,129,126)(102,168,154,132,133,120)(113,184,185,146,147,226)(115,182,183,149,145,176)(118,139,181,151,152,177)(125,197,198,158,159,172)(130,204,205,164,165,167)(134,210,202,169,170,162)(137,215,195,173,174,155)(148,231,232,187,188,269)(150,229,230,190,186,223)(153,180,228,192,193,224)(160,242,243,199,200,219)(166,249,250,206,207,214)(171,256,254,211,212,209)(175,259,247,216,217,203)(178,262,240,220,221,196)(189,265,241,225,234,276)(191,272,273,235,233,266)(194,227,271,237,238,267)(201,278,279,244,245,264)(208,282,283,251,252,261)(213,284,286,253,257,258)(218,280,285,246,260,255)(222,274,281,236,263,248)(239,270,277,268,275,287);(1,2,3)(4,5,7,10,9,8,6)(11,12,15,21,29,42,61,84,113,148,189,193,112,145,159,165,170,174,177,179,223,264,286,250,198,142,80,105,127,121,106,79,104,137,175,218,244,199,158,123,93,68,46,20,26,34,48,67,91,120,155,196,241,277,275,188,233,245,252,257,260,263,265,267,269,281,247,202,161,122,78,38,53,54,37,52,73,99,130,166,208,253,211,169,132,88,41,58,69,72,74,100,131,167,209,255,273,185,228,262,259,256,249,242,229,184,227,268,237,192,151,116,87,64,45,32,17,16,22,31,44,63,86,115,150,191,236,220,152,83,110,124,129,133,136,138,176,219,261,283,243,183,109,140,168,163,156,141,108,139,178,222,235,190,149,114,85,62,43,30,14,18,23,33,47,66,89,118,153,194,239,238,147,186,200,207,212,217,221,224,226,266,285,254,205,157,107,57,76,92,77,56,75,102,134,171,213,251,206,164,128,97,65,28,39,49,51,71,96,126,162,203,248,232,271,234,274,280,284,282,278,272,231,270,287,276,240,195,154,119,90,55,25,36,24,35,50,70,95,125,160,201,246,216,173,117,60,81,94,98,101,103,135,172,214,258,279,230,144,181,215,210,204,197,182,143,180,225,187,146,111,82,59,40,27,19,13)Polycyclic presentation:
2,3^2,7,277|c^a=c^2,d^a=d^6,e^a=e^276,d^b=d^2,e^b=e^160978 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34902, 45);