Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C5×C8×C3⋊D3 |
| Fitting | C32×C5×C8 ⋊ C2 |
| derived, Sylow | C32 ⋊ C2×C5×C8 |
| Hall | C32×C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C8 | C3⋊D3 |
| commutator, hypocenter | C32 | C2×C5×C8 |
| Frattini | C4 | C2×C5×C3⋊D3 |
| Fitting | C32×C5×C8 | C2 |
| socle | C2×C32×C5 | C2×C4 |
| prime radical | C2×C5×C3⋊D3 | C4 |
| cosocle | C32×C4 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C5×C8 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#672, #13084703, C5×#63542, … |
| maximal normal subgroup of | C3⋊D3×#672, #13084703, C5×#63542, … |
| maximal quotient of | C5×C16×C3⋊D3, C5×#63625, C5×C8×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 351);