Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×C8×D5 |
| Fitting | C32×C5×C8 ⋊ C2 |
| derived, Sylow | C5 ⋊ C2×C32×C8 |
| Hall | C32×C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C8 | D5 |
| commutator, hypocenter | C5 | C2×C32×C8 |
| Frattini | C4 | C2×C32×D5 |
| Fitting | C32×C5×C8 | C2 |
| socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C32×D5 | C4 |
| cosocle | C4×C5 | C22×C32 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C32×C8 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×D5×C3⋊C16, C3×#90934, D5×C3⋊(C3⋊C16), … |
| maximal normal subgroup of | C3×D5×C3⋊C16, C3×#90934, D5×C3⋊(C3⋊C16), … |
| maximal quotient of | C32×D5×C16, C32×#3930, C32×C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 303);