Finite Group DatabasebetaFinite Group Databasebeta| order | 59180 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic14795 |
| derived, Hall | C5×C11×C269 ⋊ C4 |
| Hall | C11×C269 ⋊ Dic5 |
| Hall | C5×C269 ⋊ Dic11 |
| Sylow | C269 ⋊ Dic55 |
| Hall | C5×C11 ⋊ Dic269 |
| Sylow | C11 ⋊ Dic1345 |
| Sylow | C5 ⋊ Dic2959 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D14795 |
| commutator, hypocenter | C5×C11×C269 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C5×C11×C269 | C2 |
| LCS | G ▷ C5×C11×C269 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 289 |
| transitive degree | μt(G) = 59180 |
A minimal faithful permutation representation — 2 generators on 289 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);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,289)(23,288)(24,287)(25,286)(26,285)(27,284)(28,283)(29,282)(30,281)(31,280)(32,279)(33,278)(34,277)(35,276)(36,275)(37,274)(38,273)(39,272)(40,271)(41,270)(42,269)(43,268)(44,267)(45,266)(46,265)(47,264)(48,263)(49,262)(50,261)(51,260)(52,259)(53,258)(54,257)(55,256)(56,255)(57,254)(58,253)(59,252)(60,251)(61,250)(62,249)(63,248)(64,247)(65,246)(66,245)(67,244)(68,243)(69,242)(70,241)(71,240)(72,239)(73,238)(74,237)(75,236)(76,235)(77,234)(78,233)(79,232)(80,231)(81,230)(82,229)(83,228)(84,227)(85,226)(86,225)(87,224)(88,223)(89,222)(90,221)(91,220)(92,219)(93,218)(94,217)(95,216)(96,215)(97,214)(98,213)(99,212)(100,211)(101,210)(102,209)(103,208)(104,207)(105,206)(106,205)(107,204)(108,203)(109,202)(110,201)(111,200)(112,199)(113,198)(114,197)(115,196)(116,195)(117,194)(118,193)(119,192)(120,191)(121,190)(122,189)(123,188)(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)(133,178)(134,177)(135,176)(136,175)(137,174)(138,173)(139,172)(140,171)(141,170)(142,169)(143,168)(144,167)(145,166)(146,165)(147,164)(148,163)(149,162)(150,161)(151,160)(152,159)(153,158)(154,157)(155,156)Polycyclic presentation:
2^2,5,11,269|a^2=b,c^a=c^4,d^a=d^10,e^a=e^26814798 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 5, 11, 269]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^268);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);