Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of A4×C5⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×C5⋊S4 |
| unambiguous semidirect product | C5⋊(A4×S4) |
| unambiguous semidirect product | A4⋊(D5×A4) |
| unambiguous semidirect product | (C22×C5⋊S4)⋊C3 |
| Fitting, socle | C24×C5 ⋊ C3×D3 |
| derived | C22×C5×A4 ⋊ C2×C3 |
| Sylow | C5 ⋊ A4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C5×A4 | C2×C3 |
| Fitting, socle | C24×C5 | C3×D3 |
| LCS | G ▷ C22×C5×A4 | |
| UCS | C1 | |
172 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 80 |
52 conjugacy classes — view the full character table.
| maximal subgroup of | D5×A4×S4, S4×C5⋊S4, #72231190, … |
| maximal normal subgroup of | D5×A4×S4, S4×C5⋊S4, #72231190, … |
| maximal quotient of | C5×A4×C5⋊S4, A4×A4⋊Dic5, #72277270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5873);