Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of (C22⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C2×C5×(C22⋊C9)⋊C2 |
| unambiguous semidirect product | (C22⋊C9)⋊(C22×C5) |
| unambiguous semidirect product | (C2×C22⋊C9)⋊(C2×C5) |
| derived | C22⋊C9 ⋊ C22×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | (C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | C22⋊C9 | C22×C5 |
| Frattini | C3 | C2×C5×S4 |
| Fitting, socle | C23×C3×C5 | D3 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C2×C5 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 396);