Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D32×D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | D32×Dic5 |
| derived, Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C22×Dic5 |
| Sylow | C5 ⋊ C4×D32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×D5 |
| commutator, hypocenter | C32×C5 | C22×C4 |
| Fitting, socle, cosocle | C2×C32×C5 | C23 |
| prime radical | C2×C5×D32 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | D32⋊#170, D32×#170, D3×#6739, … |
| maximal normal subgroup of | D32⋊#170, D32×#170, D3×#6739, … |
| maximal quotient of | D32×#168, D3×#6650, #13085645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 489);