Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×D3×Dic3 |
| derived, Sylow | C32 ⋊ C22×C4×C5 |
| Hall | C32×C5 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | D32 |
| commutator, hypocenter | C32 | C22×C4×C5 |
| Frattini | C2 | C2×C5×D32 |
| Fitting, socle | C22×C32×C5 | C22 |
| prime radical | C22×C3×C5×D3 | C2 |
| cosocle | C2×C32 | C23×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C5 | |
232 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
180 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic3×Dic5, D3×#6665, D3×#6667, … |
| maximal normal subgroup of | D3×Dic3×Dic5, D3×#6665, D3×#6667, … |
| maximal quotient of | C5×Dic3×Dic6, C5×#63843, C5×#63855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 652);