Finite Group DatabasebetaFinite Group Databasebeta| order | 20510 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D10255 |
| unambiguous semidirect product | C5⋊D2051 |
| unambiguous semidirect product | C7⋊D1465 |
| unambiguous semidirect product | C293⋊D35 |
| Fitting, socle, derived, Hall | C5×C7×C293 ⋊ C2 |
| Hall | C7×C293 ⋊ D5 |
| Hall | C5×C293 ⋊ D7 |
| Sylow | C293 ⋊ D35 |
| Hall | C5×C7 ⋊ D293 |
| Sylow | C7 ⋊ D1465 |
| Sylow | C5 ⋊ D2051 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C7×C293 | C2 |
| LCS | G ▷ C5×C7×C293 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 305 |
| transitive degree | μt(G) = 10255 |
A minimal faithful permutation representation — 2 generators on 305 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);(2,5)(3,4)(7,12)(8,11)(9,10)(14,305)(15,304)(16,303)(17,302)(18,301)(19,300)(20,299)(21,298)(22,297)(23,296)(24,295)(25,294)(26,293)(27,292)(28,291)(29,290)(30,289)(31,288)(32,287)(33,286)(34,285)(35,284)(36,283)(37,282)(38,281)(39,280)(40,279)(41,278)(42,277)(43,276)(44,275)(45,274)(46,273)(47,272)(48,271)(49,270)(50,269)(51,268)(52,267)(53,266)(54,265)(55,264)(56,263)(57,262)(58,261)(59,260)(60,259)(61,258)(62,257)(63,256)(64,255)(65,254)(66,253)(67,252)(68,251)(69,250)(70,249)(71,248)(72,247)(73,246)(74,245)(75,244)(76,243)(77,242)(78,241)(79,240)(80,239)(81,238)(82,237)(83,236)(84,235)(85,234)(86,233)(87,232)(88,231)(89,230)(90,229)(91,228)(92,227)(93,226)(94,225)(95,224)(96,223)(97,222)(98,221)(99,220)(100,219)(101,218)(102,217)(103,216)(104,215)(105,214)(106,213)(107,212)(108,211)(109,210)(110,209)(111,208)(112,207)(113,206)(114,205)(115,204)(116,203)(117,202)(118,201)(119,200)(120,199)(121,198)(122,197)(123,196)(124,195)(125,194)(126,193)(127,192)(128,191)(129,190)(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)(136,183)(137,182)(138,181)(139,180)(140,179)(141,178)(142,177)(143,176)(144,175)(145,174)(146,173)(147,172)(148,171)(149,170)(150,169)(151,168)(152,167)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)Polycyclic presentation:
2,5,7,293|b^a=b^4,c^a=c^6,d^a=d^2925129 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20510, 8);