Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| direct product | Dic3×SL(2,𝔽5) |
| unambiguous semidirect product | C3⋊(C4×SL(2,𝔽5)) |
| derived | C3×SL(2,𝔽5) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×A5 |
| commutator, hypocenter | C3×SL(2,𝔽5) | C4 |
| Fitting, socle, cosocle | C22×C3 | C2×A5 |
| prime radical | C2×C3×SL(2,𝔽5) | C2 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C22 | |
120 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽5)2, #226×SL(2,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 1690);