Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C3×C4 |
| direct product | C4×C5×C3⋊Dic3 |
| derived, Sylow | C32 ⋊ C42×C5 |
| Hall | C32×C5 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C5 | C3⋊D3 |
| commutator, hypocenter | C32 | C42×C5 |
| Frattini | C22 | C2×C5×C3⋊D3 |
| Fitting | C2×C32×C4×C5 | C2 |
| socle, prime radical | C22×C32×C5 | C22 |
| cosocle | C22×C32 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C4×C5 | |
180 conjugacy classes of subgroups.
| transitive degree | μt(G) = 720 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊Dic3×#168, #13084721, #13084745, … |
| maximal normal subgroup of | C3⋊Dic3×#168, #13084721, #13084745, … |
| maximal quotient of | C4×C5×C3⋊(C3⋊C8), C5×#63630, C5×C8×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 358);