Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of A4⋊S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | A4⋊S5 |
| unambiguous semidirect product | A5⋊S4 |
| Fitting | C22 ⋊ ΓL(2,𝔽4) |
| socle | C22×A5 ⋊ D3 |
| derived | A4×A5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×A5 | C2 |
| Fitting | C22 | ΓL(2,𝔽4) |
| socle | C22×A5 | D3 |
| LCS | G ▷ A4×A5 | |
| UCS | C1 | |
143 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 20 |
Permutation generators:
(1,3,4,2)(6,9,8,7);(1,5,2,4,3)(6,8,7)19 conjugacy classes — view the full character table.
| maximal subgroup of | #72219548, S4×S5, C2×A4⋊S5, … |
| maximal normal subgroup of | S4×S5, C2×A4⋊S5, C3×A4⋊S5 |
| maximal quotient of | #72219544, C2×A4⋊S5, #72240863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5847);