Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#64210 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D3×S4 |
| commutator, cosocle | C2×C3×A4 | C22×C5 |
| Frattini | C2 | C5×D3×S4 |
| Fitting, socle | C23×C3×C5 | C2×D3 |
| hypocenter | C3×A4 | C5×D4 |
| LCS | G ▷ C2×C3×A4 ▷ C3×A4 | |
| UCS | C1 ◁ C2×C5 | |
268 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 180 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,4,5)(6,7,8,9)(10,11,13,12)(14,15,16);(7,9)(11,12,13)(15,16)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4847);