Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C3⋊D15 |
| Fitting, socle | C23×C32×C5 ⋊ C2 |
| derived, Hall | C32×C5 ⋊ C24 |
| Sylow | C32 ⋊ C23×D5 |
| Sylow | C5 ⋊ C23×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊D15 |
| commutator, cosocle, hypocenter | C32×C5 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32×C5 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C23 | |
804 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10623772, #13087039, #13087052, … |
| maximal normal subgroup of | C2×#10623772, #13087039, #13087052, … |
| maximal quotient of | C22×#67180, C22×C4×C3⋊D15, C22×C4⋊(C3⋊D15), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 839);