Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊(C5⋊GL(2,𝔽3)) |
| unambiguous semidirect product | (C2×C3)⋊(C5⋊GL(2,𝔽3)) |
| unambiguous semidirect product | (C2×C5)⋊(C3⋊GL(2,𝔽3)) |
| Fitting | C2×C3×C5×Q8 ⋊ D3 |
| derived | C3×C5×SL(2,𝔽3) ⋊ C22 |
| Sylow | C5 ⋊ C2×C3⋊GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C3×C5×SL(2,𝔽3) | C22 |
| Frattini | C2 | C2×C3⋊(C5⋊S4) |
| Fitting | C2×C3×C5×Q8 | D3 |
| socle | C22×C3×C5 | S4 |
| LCS | G ▷ C3×C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
288 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
114 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72229353, C2×D5×C3⋊GL(2,𝔽3) |
| maximal normal subgroup of | C2×D5×C3⋊GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5399);