Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×Dic15 |
| unambiguous semidirect product | C3⋊(Dic5×SL(2,𝔽3)) |
| unambiguous semidirect product | (Q8×Dic15)⋊C3 |
| derived | C3×C5×Q8 ⋊ C3×C4 |
| Sylow | C5 ⋊ Dic3×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | A4×D15 |
| commutator, hypocenter | C3×C5×Q8 | C3×C4 |
| Fitting, cosocle | C2×C3×C5×Q8 | C2×C3 |
| socle | C22×C3×C5 | C2×A4 |
| prime radical | C2×C3×C5×SL(2,𝔽3) | C2 |
| LCS | G ▷ C3×C5×Q8 | |
| UCS | C1 ◁ C22 | |
134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 480 |
126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2555);