Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | D4×C3⋊D15 |
| Fitting | C32×C5×D4 ⋊ C2 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ D4×D5 |
| Sylow | C5 ⋊ D4×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C3⋊D15 |
| commutator, socle, cosocle | C2×C32×C5 | C23 |
| Fitting | C32×C5×D4 | C2 |
| hypercenter | D4 | C3⋊D15 |
| hypocenter | C32×C5 | C2×D4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ D4 | |
324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #13086385, #13086964, #13086970, … |
| maximal normal subgroup of | #13086385, #13086964, #13086970, … |
| maximal quotient of | #13088147, #13088148, C22⋊C4×C3⋊D15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 753);