Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C32×Dic5 |
| derived, Sylow | C5 ⋊ C22×C32×C4 |
| Hall | C32×C5 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C32 | D5 |
| commutator, hypocenter | C5 | C22×C32×C4 |
| Frattini | C2 | C22×C32×D5 |
| Fitting, socle, prime radical | C23×C32×C5 | C2 |
| cosocle | C2×C5 | C23×C32 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23×C32 | |
324 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#90994, #13084762, C32×#3963, … |
| maximal normal subgroup of | C3×#90994, #13084762, C32×#3963, … |
| maximal quotient of | C22×C32×#168, C2×C32×#681, C2×C32×C4×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 717);