Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊#6728 |
| unambiguous semidirect product | (C2×C3)⋊#6728 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ΓL(2,𝔽4) |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽5) | C22 |
| Frattini | C2 | C2×ΓL(2,𝔽4) |
| Fitting, socle | C22×C3 | S5 |
| prime radical | C2×C3×SL(2,𝔽5) | C2 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C22 | |
150 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4637);