Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C3×C4 |
| direct product | C32×C4×Dic5 |
| derived, Sylow | C5 ⋊ C32×C42 |
| Hall | C32×C5 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32×C4 | D5 |
| commutator, hypocenter | C5 | C32×C42 |
| Frattini | C22 | C2×C32×D5 |
| Fitting | C2×C32×C4×C5 | C2 |
| socle, prime radical | C22×C32×C5 | C22 |
| cosocle | C22×C5 | C22×C32 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C32×C4 | |
180 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×Dic5×C3⋊C8, C3×#90952, C3×#90979, … |
| maximal normal subgroup of | C3×Dic5×C3⋊C8, C3×#90952, C3×#90979, … |
| maximal quotient of | C32×C4×#168, C32×#3935, C32×C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 310);