Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | D5×Q8⋊C9 |
| unambiguous semidirect product | C5⋊(C2×Q8⋊C9) |
| unambiguous semidirect product | (Q8×D5)⋊C9 |
| derived | C5×Q8 ⋊ C2×C9 |
| Hall | Q8×D5 ⋊ C9 |
| Sylow | C5 ⋊ C2×Q8⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | D5×A4 |
| commutator, hypocenter | C5×Q8 | C2×C9 |
| Fitting, cosocle | C3×C5×Q8 | C2×C3 |
| socle | C2×C3×C5 | C2×A4 |
| prime radical | C2×C3×D5 | A4 |
| LCS | G ▷ C5×Q8 | |
| UCS | C1 ◁ C2×C3 | |
58 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | #13085141, AGL(1,𝔽5)×Q8⋊C9, D5×#3688, … |
| maximal normal subgroup of | #13085141, AGL(1,𝔽5)×Q8⋊C9, D5×#3688, … |
| maximal quotient of | Dic5×Q8⋊C9, C2×D5×Q8⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 109);