Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#63715 |
| Fitting | C5×C9×#41 ⋊ C2 |
| Hall | C5×C9 ⋊ H(5,𝔽2) |
| Sylow | C9 ⋊ C5×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C23×D9 |
| commutator, cosocle | C2×C9 | C24×C5 |
| Frattini | C2×C3 | C23×C5×D3 |
| Fitting | C5×C9×#41 | C2 |
| socle | C2×C3×C5 | C23×D3 |
| hypercenter | C5×#41 | D9 |
| hypocenter | C9 | C5×H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C5 ◁ C5×#41 | |
498 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3,4,5)(6,11,13,12,7,9,8,14,10)(15,17,18,21)(16,20,19,22);(16,19)(20,22);(15,16)(17,20)(18,19)(21,22);(7,9)(8,12)(10,11)(13,14)(17,21)(20,22)285 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 1657);