Finite Group DatabasebetaFinite Group Databasebeta| order | 34870 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D17435 |
| unambiguous semidirect product | C5⋊D3487 |
| unambiguous semidirect product | C11⋊D1585 |
| unambiguous semidirect product | C317⋊D55 |
| Fitting, socle, derived, Hall | C5×C11×C317 ⋊ C2 |
| Hall | C11×C317 ⋊ D5 |
| Hall | C5×C317 ⋊ D11 |
| Sylow | C317 ⋊ D55 |
| Hall | C5×C11 ⋊ D317 |
| Sylow | C11 ⋊ D1585 |
| Sylow | C5 ⋊ D3487 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C11×C317 | C2 |
| LCS | G ▷ C5×C11×C317 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 333 |
| transitive degree | μt(G) = 17435 |
A minimal faithful permutation representation — 2 generators on 333 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333);(2,5)(3,4)(7,16)(8,15)(9,14)(10,13)(11,12)(18,333)(19,332)(20,331)(21,330)(22,329)(23,328)(24,327)(25,326)(26,325)(27,324)(28,323)(29,322)(30,321)(31,320)(32,319)(33,318)(34,317)(35,316)(36,315)(37,314)(38,313)(39,312)(40,311)(41,310)(42,309)(43,308)(44,307)(45,306)(46,305)(47,304)(48,303)(49,302)(50,301)(51,300)(52,299)(53,298)(54,297)(55,296)(56,295)(57,294)(58,293)(59,292)(60,291)(61,290)(62,289)(63,288)(64,287)(65,286)(66,285)(67,284)(68,283)(69,282)(70,281)(71,280)(72,279)(73,278)(74,277)(75,276)(76,275)(77,274)(78,273)(79,272)(80,271)(81,270)(82,269)(83,268)(84,267)(85,266)(86,265)(87,264)(88,263)(89,262)(90,261)(91,260)(92,259)(93,258)(94,257)(95,256)(96,255)(97,254)(98,253)(99,252)(100,251)(101,250)(102,249)(103,248)(104,247)(105,246)(106,245)(107,244)(108,243)(109,242)(110,241)(111,240)(112,239)(113,238)(114,237)(115,236)(116,235)(117,234)(118,233)(119,232)(120,231)(121,230)(122,229)(123,228)(124,227)(125,226)(126,225)(127,224)(128,223)(129,222)(130,221)(131,220)(132,219)(133,218)(134,217)(135,216)(136,215)(137,214)(138,213)(139,212)(140,211)(141,210)(142,209)(143,208)(144,207)(145,206)(146,205)(147,204)(148,203)(149,202)(150,201)(151,200)(152,199)(153,198)(154,197)(155,196)(156,195)(157,194)(158,193)(159,192)(160,191)(161,190)(162,189)(163,188)(164,187)(165,186)(166,185)(167,184)(168,183)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)Polycyclic presentation:
2,5,11,317|b^a=b^4,c^a=c^10,d^a=d^3168719 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34870, 9);