Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×C5×S4 |
| unambiguous semidirect product | A4⋊(C22×C3×C5) |
| unambiguous semidirect product | (C2×A4)⋊(C2×C3×C5) |
| Fitting, socle | C23×C3×C5 ⋊ D3 |
| derived | A4 ⋊ C22×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | S4 |
| commutator, cosocle, hypocenter | A4 | C22×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C23×C3×C5 | D3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C3×C5 | |
140 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 90 |
150 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 814);