Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of S4×A5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| direct product | S4×A5 |
| unambiguous semidirect product | A4⋊(C2×A5) |
| Fitting | C22 ⋊ D3×A5 |
| socle | C22×A5 ⋊ D3 |
| derived | A4×A5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | A4×A5 | C2 |
| Fitting | C22 | D3×A5 |
| socle | C22×A5 | D3 |
| cosocle | A4 | C2×A5 |
| LCS | G ▷ A4×A5 | |
| UCS | C1 | |
154 conjugacy classes of subgroups; only the 8 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 8 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 20 |
25 conjugacy classes — view the full character table.
| maximal subgroup of | A5×GL(3,𝔽2), A5×A6, S4×S5, … |
| maximal normal subgroup of | S4×S5, C2×S4×A5, C3×S4×A5 |
| maximal quotient of | #226×A5, S4×SL(2,𝔽5), #72209752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5848);