Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #1201 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1203 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C22⋊D5 |
| Sylow | C5 ⋊ C2×C3×D12 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D3×D5 |
| commutator, cosocle | C2×C3×C5 | C23×C3 |
| Frattini | C2 | C2×C3×D3×D5 |
| Fitting, socle | C22×C32×C5 | C22 |
| hypocenter | C3×C5 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22×C3 | |
232 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#91174, #13085469, Dic3×#1203, … |
| maximal normal subgroup of | C3×#91174, #13085469, Dic3×#1203, … |
| maximal quotient of | C3×#91373, C3×#91375, C3×#91376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 611);