Finite Group DatabasebetaFinite Group Databasebeta| order | 32490 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20266582 |
| derived, Hall | C5×C192 ⋊ C2×C9 |
| Hall | C5×#15692429 ⋊ C2 |
| Hall | C5⋊(C19⋊D19) ⋊ C9 |
| Hall | #15692429 ⋊ D5 |
| Sylow | C192 ⋊ C9×D5 |
| Sylow | C5 ⋊ #20266582 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C5⋊#14524206 |
| commutator, hypocenter | C5×C192 | C2×C9 |
| Fitting, socle, cosocle | C3×C5×C192 | C2×C3 |
| prime radical | C3×C5⋊(C19⋊D19) | C3 |
| LCS | G ▷ C5×C192 | |
| UCS | C1 ◁ C3 | |
264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 16245 |
A minimal faithful permutation representation — 3 generators on 52 points:
(2,3)(4,5)(6,10,14,8,13,9,11,7,12)(16,20,21,17,18,22)(19,28,25,23,24,30)(26,31,33,32,29,27)(35,39,40,36,37,41)(38,47,44,42,43,49)(45,50,52,51,48,46);(1,2,4,5,3)(34,35,38,44,50,47,51,37,41,46,52,40,39,45,43,48,49,42,36);(15,16,19,25,31,28,32,18,22,27,33,21,20,26,24,29,30,23,17)Polycyclic presentation:
2,3,5,19^2,3|b^3=f,c^a=c^4,d^a=d^18,e^a=e^18,d^b=d^7,e^b=e^7936 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32490, 56);