Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×#168 |
| derived, Sylow | C5 ⋊ C2×C8×C9 |
| Hall | C5×C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C9 | D5 |
| commutator, hypocenter | C5 | C2×C8×C9 |
| Frattini | C3×C4 | C2×C3×D5 |
| Fitting | C2×C4×C5×C9 | C2 |
| socle, prime radical | C22×C3×C5 | C3×C4 |
| cosocle | C3×C4×C5 | C22×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C4×C9 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | Dic9×#168, #13083722, #13083723, … |
| maximal normal subgroup of | Dic9×#168, #13083722, #13083723, … |
| maximal quotient of | C4×C9×#168, C9×C4⋊#168, C2×C9×#672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 38);