Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of A4×S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×S5 |
| unambiguous semidirect product | A5⋊(C2×A4) |
| unambiguous semidirect product | (C22×S5)⋊C3 |
| Fitting | C22 ⋊ C3×S5 |
| socle, derived | C22×A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C22×A5 | C2×C3 |
| Fitting | C22 | C3×S5 |
| LCS | G ▷ C22×A5 | |
| UCS | C1 | |
146 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 20 |
28 conjugacy classes — view the full character table.
| maximal subgroup of | S4×S5, A5×S5, C2×A4×S5 |
| maximal normal subgroup of | S4×S5, C2×A4×S5 |
| maximal quotient of | A4×A5⋊C4, SL(2,𝔽3)×S5, #72276735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5846);