Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×S5 |
| unambiguous semidirect product | A5⋊(C22×C3) |
| unambiguous semidirect product | (C2×A5)⋊(C2×C3) |
| socle | C2×C3×A5 ⋊ C2 |
| derived | A5 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3 | S5 |
| commutator, cosocle, hypocenter | A5 | C22×C3 |
| Frattini | C1 | G |
| socle | C2×C3×A5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3 | |
121 conjugacy classes of subgroups.
| transitive degree | μt(G) = 30 |
42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 769);