Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#4150 |
| Fitting | C32×C5×#41 ⋊ C2 |
| Hall | C32×C5 ⋊ 2−1+4 |
| Sylow | C5 ⋊ C32×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C23×D5 |
| commutator, cosocle | C2×C5 | C24×C32 |
| Frattini | C2 | C23×C32×D5 |
| Fitting | C32×C5×#41 | C2 |
| socle | C2×C32×C5 | C24 |
| hypercenter | C32×#41 | D5 |
| hypocenter | C5 | C32×2−1+4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C32 ◁ C32×#41 | |
876 conjugacy classes of subgroups; only the 510 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 510 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(7,8,10,11,9)(12,16,17,15)(13,19,20,24)(14,21,22,23)(18,26,27,25);(4,5,6)(12,15,17,16)(13,19,20,24)(14,21,22,23)(18,25,27,26);(12,14,17,22)(13,18,20,27)(15,23,16,21)(19,26,24,25);(8,9)(10,11)(12,13)(14,18)(15,19)(16,24)(17,20)(21,25)(22,27)(23,26)333 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5772);