Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C3⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | C22×C3⋊(C5⋊S4) |
| Fitting, socle | C24×C3×C5 ⋊ D3 |
| derived | C3×C5×A4 ⋊ C23 |
| Sylow | C5 ⋊ C22×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C3×C5×A4 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C22 | |
652 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 652 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | C22×D5×C3⋊S4, C22×C5⋊(A4⋊S4) |
| maximal normal subgroup of | C22×D5×C3⋊S4 |
| maximal quotient of | C22×C5⋊(A4⋊S4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5928);