Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of GO+(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×GO+(4,𝔽2) |
| unambiguous semidirect product | #157⋊(C22×C5) |
| unambiguous semidirect product | D32⋊(C22×C5) |
| Fitting, socle | C2×C32×C5 ⋊ D4 |
| Hall | C32×C5 ⋊ C2×D4 |
| Sylow | C32 ⋊ C2×C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | GO+(4,𝔽2) |
| commutator, cosocle | C3⋊D3 | C23×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | D4 |
| hypocenter | C32 | C2×C5×D4 |
| LCS | G ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
172 conjugacy classes of subgroups.
| transitive degree | μt(G) = 60 |
90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 808);