Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of A4×A5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×A4×A5 |
| unambiguous semidirect product | C23⋊(C3×A5) |
| unambiguous semidirect product | (C23×A5)⋊C3 |
| Fitting | C23 ⋊ C3×A5 |
| socle | C23×A5 ⋊ C3 |
| derived | C22×A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×A5 |
| commutator, hypocenter | C22×A5 | C2×C3 |
| Frattini | C1 | G |
| Fitting | C23 | C3×A5 |
| socle | C23×A5 | C3 |
| cosocle | C22 | C2×C3×A5 |
| LCS | G ▷ C22×A5 | |
| UCS | C1 ◁ C2 | |
196 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 30 |
40 conjugacy classes — view the full character table.
| derived subgroup of | A5×GL(2,ℤ/4) |
| maximal subgroup of | A5×#997, C22×A4×A5, A5×#120⋊C3, … |
| maximal normal subgroup of | C22×A4×A5, C2×S4×A5, SL(2,ℤ/4)×A5, … |
| maximal quotient of | C22×A4×A5, A5×#120⋊C3, D3×A4×A5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5850);