Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C16 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×C3⋊C16 |
| unambiguous semidirect product | C3⋊(C3×C5×C16) |
| derived | C3 ⋊ C3×C5×C16 |
| Hall | C32×C5 ⋊ C16 |
| Sylow | C32 ⋊ C5×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5×C8 | D3 |
| commutator, hypocenter | C3 | C3×C5×C16 |
| Frattini | C8 | C3×C5×D3 |
| Fitting | C32×C5×C8 | C2 |
| socle, prime radical | C2×C32×C5 | C8 |
| cosocle | C3×C8 | C2×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C5×C8 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
360 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×D5×C3⋊C16, C3×#90933, C3×#90934, … |
| maximal normal subgroup of | C3×D5×C3⋊C16, C3×#90933, C3×#90934, … |
| maximal quotient of | C3×C5×C3⋊C32, C2×C3×C5×C3⋊C16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 90);