Finite Group DatabasebetaFinite Group Databasebeta| order | 3420 |
| isoclinism | isoclinic class of #4379 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C19 · Schur multiplier (H2) = C3 |
| direct product | C19×#4379 |
| Fitting, socle, Hall | C32×C5×C19 ⋊ C4 |
| derived, Hall | C32×C5 ⋊ C4×C19 |
| Hall | C32×C19 ⋊ Dic5 |
| Hall | C5×C19 ⋊ #157 |
| Sylow | C32 ⋊ C19×Dic5 |
| Sylow | C5 ⋊ C19×#157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C19 | #4379 |
| commutator, hypocenter | C32×C5 | C4×C19 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C19 | C4 |
| prime radical | C5×C3⋊D3×C19 | C2 |
| cosocle | C5×C3⋊D3 | C2×C19 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C19 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 570 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23,25,24)(27,28)(29,30);(20,22,25)(26,27,29,30,28)342 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C19×#11744620, C5×C19×#4379, C19×#11744709, … |
| maximal normal subgroup of | C5×C19×#4379, C7×C19×#4379, C11×C19×#4379 |
| maximal quotient of | C19×#11744620, C5×C19×#4379, C19×#11744709, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3420, 126);