Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D3×#1220 |
| Fitting | C22×C32×C5 ⋊ C22 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C22⋊D5 |
| Sylow | C5 ⋊ D3×#67 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D15 |
| commutator, socle, cosocle | C2×C32×C5 | C23 |
| Fitting | C22×C32×C5 | C22 |
| hypercenter | C22 | D3×D15 |
| hypocenter | C32×C5 | C2×D4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
232 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
99 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6779, D3×#6782, #13088989, … |
| maximal normal subgroup of | D3×#6779, D3×#6782, #13088989, … |
| maximal quotient of | D3×#6712, #13087406, #13087411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 686);