Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3 |
| direct product | C23×C32×D5 |
| unambiguous semidirect product | C5⋊(C24×C32) |
| unambiguous semidirect product | (C2×C5)⋊(C23×C32) |
| Fitting, socle | C23×C32×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C24×C32 |
| Hall | C32×C5 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C32 | D5 |
| commutator, cosocle, hypocenter | C5 | C24×C32 |
| Frattini | C1 | G |
| Fitting, socle | C23×C32×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C32 | |
804 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C3×#6665, C3×D5×#217, C3×#91563, … |
| maximal normal subgroup of | C2×C3×#6665, C3×D5×#217, C3×#91563, … |
| maximal quotient of | C22×C32×Dic10, C22×C32×C4×D5, C22×C32×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 835);