Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C24 |
| direct product | C22×C3×C5×S4 |
| unambiguous semidirect product | A4⋊(C23×C3×C5) |
| unambiguous semidirect product | (C2×A4)⋊(C22×C3×C5) |
| Fitting, socle | C24×C3×C5 ⋊ D3 |
| derived | A4 ⋊ C23×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C5 | S4 |
| commutator, cosocle, hypocenter | A4 | C23×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C3×C5 | |
544 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C32×C5×S4, C22×C3×C5×GL(3,𝔽2) |
| maximal normal subgroup of | C22×C32×C5×S4 |
| maximal quotient of | C22×C32×C5×S4, C22×C3×C5×GL(2,𝔽3) |
| automorphism group of | #41×C31 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5925);