Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C23×C3×D15 |
| Fitting, socle | C23×C32×C5 ⋊ C2 |
| derived | C3×C5 ⋊ C24×C3 |
| Hall | C32×C5 ⋊ C24 |
| Sylow | C32 ⋊ C23×D5 |
| Sylow | C5 ⋊ C23×C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D15 |
| commutator, cosocle, hypocenter | C3×C5 | C24×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32×C5 | C2 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C23×C3 | |
568 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×#6667, C3×#91561, C3×#91572, … |
| maximal normal subgroup of | C2×C3×#6667, C3×#91561, C3×#91572, … |
| maximal quotient of | C22×C3×Dic30, C22×C3×C4×D15, C22×C3×D60, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 837);