Finite Group DatabasebetaFinite Group Databasebeta| order | 26586 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#22579727 |
| Fitting, socle, derived | C3×C7×C211 ⋊ C2×C3 |
| Hall | C3×#17984522 ⋊ C2 |
| Hall | C7×C211 ⋊ C3×D3 |
| Sylow | C211 ⋊ C3⋊AGL(1,𝔽7) |
| Sylow | C7 ⋊ D3×C211⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C7×C211 | C2×C3 |
| LCS | G ▷ C3×C7×C211 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 4431 |
A minimal faithful permutation representation — 2 generators on 221 points:
(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,35)(18,27,32)(21,30,29)(23,33,26)(25,37,51)(28,40,48)(31,43,45)(34,46,42)(36,49,39)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,144)(98,122,141)(101,125,138)(104,128,135)(107,131,133)(110,120,130)(113,136,127)(116,139,124)(118,142,121)(123,147,168)(126,150,165)(129,153,162)(132,156,160)(134,148,158)(137,161,155)(140,163,152)(143,166,149)(145,169,146)(151,172,187)(154,175,184)(157,178,182)(159,173,181)(164,183,177)(167,185,174)(170,188,171)(176,191,200)(179,194,198)(180,192,197)(186,199,193)(189,201,190)(195,204,208)(196,205,207)(202,209,203)(206,210,211)(213,216,217,214,215,218)(220,221);(1,2,4,8,15,25,38,54,73,95,120,84,46,20,5,9,16,26,39,55,74,96,121,146,161,109,65,33,12,19,29,42,58,77,99,124,149,171,183,136,87,49,22,32,45,61,80,102,127,152,174,190,199,163,112,68,35,48,64,83,105,130,155,177,193,203,209,185,139,90,51,67,86,108,133,158,119,145,170,189,202,201,166,115,70,89,111,135,160,181,147,94,118,143,167,186,188,142,92,114,138,162,182,197,172,122,72,93,116,140,164,169,117,141,165,184,198,207,191,150,97,53,71,91,113,137,144,168,187,200,208,211,204,175,125,75,37,52,69,88,110,134,159,180,196,206,210,194,153,100,56,24,36,50,66,85,107,132,157,179,195,205,178,128,78,40,14,23,34,47,63,82,104,129,154,176,192,156,103,59,27,7,13,21,31,44,60,79,101,126,151,173,131,81,43,17,3,6,11,18,28,41,57,76,98,123,148,106,62,30,10)(212,213,215,218,217,216,214)(219,220,221)Polycyclic presentation:
2,3^2,7,211|c^a=c^2,d^a=d^6,d^b=d^2,e^b=e^196852 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26586, 64);