Finite Group DatabasebetaFinite Group Databasebeta| order | 20090 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | C5⋊(C7⋊D287) |
| unambiguous semidirect product | C7⋊D1435 |
| unambiguous semidirect product | C41⋊(C5⋊(C7⋊D7)) |
| Fitting, socle, derived, Hall | C5×C72×C41 ⋊ C2 |
| Hall | C72×C41 ⋊ D5 |
| Hall | C5×C72 ⋊ D41 |
| Hall | C5×C41 ⋊ C7⋊D7 |
| Sylow | C72 ⋊ D205 |
| Sylow | C41 ⋊ C5⋊(C7⋊D7) |
| Sylow | C5 ⋊ C7⋊D287 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C72×C41 | C2 |
| LCS | G ▷ C5×C72×C41 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 60 |
| transitive degree | μt(G) = 10045 |
A minimal faithful permutation representation — 3 generators on 60 points:
(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)(43,44)(45,46)(48,49)(50,51)(52,53)(55,56)(57,58)(59,60);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(42,43,45,46,44)(54,55,57,59,60,58,56);(47,48,50,52,53,51,49)Polycyclic presentation:
2,5,7^2,41|b^a=b^4,c^a=c^6,d^a=d^6,e^a=e^405024 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20090, 30);