Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D32⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D32⋊D5 |
| Fitting, socle | C2×C32×C5 ⋊ D4 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C22⋊D5 |
| Sylow | C5 ⋊ C2×GO+(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32⋊D5 |
| commutator, cosocle | C5×C3⋊D3 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | D4 |
| hypocenter | C32×C5 | C2×D4 |
| LCS | G ▷ C5×C3⋊D3 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
172 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 60 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | #13088444, #13088446, #13088447, … |
| maximal normal subgroup of | #13088444, #13088446, #13088447, … |
| maximal quotient of | #13088826, #13088827, #13088828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 809);