Finite Group DatabasebetaFinite Group Databasebeta| order | 30420 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C13 |
| direct product | C3×D5×#11759816 |
| Fitting, socle | C3×C5×C132 ⋊ C22×C3 |
| derived, Hall | C5×C132 ⋊ C22×C32 |
| Hall | C3×C5×#92789 ⋊ C22 |
| Hall | D5×C13⋊D13 ⋊ C32 |
| Hall | C3×#92789 ⋊ C2×D5 |
| Sylow | C132 ⋊ C2×C32×D5 |
| Sylow | C5 ⋊ C2×C3×#11759816 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D5×#11759816 |
| commutator, cosocle, hypocenter | C5×C132 | C22×C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C132 | C22×C3 |
| LCS | G ▷ C5×C132 | |
| UCS | C1 ◁ C3 | |
320 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 585 |
A minimal faithful permutation representation — 2 generators on 34 points:
(4,5,7,8,6)(10,14,15,11,12,16)(13,20,19,17,18,21)(23,27,28,24,25,29)(26,32,33,30,31,34);(1,2,3)(5,6)(7,8)(9,10,13,12,16,19,18,20,21,15,14,17,11)(22,23,26,28,27,31,33,34,32,25,29,30,24)408 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30420, 581);