Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C3⋊(#120⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×C3⋊(#120⋊C3) |
| unambiguous semidirect product | C3⋊(C5×#120⋊C3) |
| unambiguous semidirect product | #953⋊(C3×C5) |
| unambiguous semidirect product | (C5×#953)⋊C3 |
| Fitting | C24×C3×C5 ⋊ C2×C3 |
| derived | C24×C3 ⋊ C2×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊(#120⋊C3) |
| commutator, cosocle, hypocenter | C24×C3 | C2×C3×C5 |
| Frattini | C22 | C5×D3×A4 |
| Fitting | C24×C3×C5 | C2×C3 |
| socle | C22×C3×C5 | C2×A4 |
| LCS | G ▷ C24×C3 | |
| UCS | C1 ◁ C5 | |
154 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4,5)(7,8)(9,10)(11,14,16,15,12,13);(6,7,8)(10,15)(13,14)90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2550);