Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×C4×Dic15 |
| derived | C3×C5 ⋊ C3×C42 |
| Hall | C32×C5 ⋊ C42 |
| Sylow | C32 ⋊ C4×Dic5 |
| Sylow | C5 ⋊ C3×C4×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D15 |
| commutator, hypocenter | C3×C5 | C3×C42 |
| Frattini | C22 | C2×C3×D15 |
| Fitting | C2×C32×C4×C5 | C2 |
| socle, prime radical | C22×C32×C5 | C22 |
| cosocle | C22×C3×C5 | C22×C3 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C3×C4 | |
136 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#90951, C3×#90954, C3×#90976, … |
| maximal normal subgroup of | C3×#90951, C3×#90954, C3×#90976, … |
| maximal quotient of | C3×C4×C3⋊#168, C3×#91087, C3×C8×Dic15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 342);