Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C3⋊D3×AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C3×C4 |
| direct product | AGL(1,𝔽5)×C3⋊Dic3 |
| unambiguous semidirect product | (C3×AGL(1,𝔽5))⋊Dic3 |
| derived, Hall | C32×C5 ⋊ C42 |
| Sylow | C32 ⋊ C4×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×C3⋊Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊D3×AGL(1,𝔽5) |
| commutator, hypocenter | C32×C5 | C42 |
| Fitting, socle | C2×C32×C5 | C2×C4 |
| prime radical, cosocle | C2×C32×D5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
180 conjugacy classes of subgroups.
| transitive degree | μt(G) = 180 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 432);