Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of AGL(1,𝔽5)×C3⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | AGL(1,𝔽5)×C3⋊S4 |
| unambiguous semidirect product | (C3×AGL(1,𝔽5))⋊S4 |
| unambiguous semidirect product | (A4×AGL(1,𝔽5))⋊D3 |
| Fitting, socle | C22×C3×C5 ⋊ C4×D3 |
| derived | C3×C5×A4 ⋊ C2×C4 |
| Sylow | C5 ⋊ C4×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | C3×C5×A4 | C2×C4 |
| Fitting, socle | C22×C3×C5 | C4×D3 |
| prime radical | D5×C3⋊S4 | C2 |
| cosocle | C3×D5×A4 | C22 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 | |
288 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 5866);