Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4×AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽5)×C22⋊C9 |
| unambiguous semidirect product | (C22×AGL(1,𝔽5))⋊C9 |
| derived | C22×C5 ⋊ C4×C9 |
| Hall | C22×AGL(1,𝔽5) ⋊ C9 |
| Sylow | C5 ⋊ C4×C22⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A4×AGL(1,𝔽5) |
| commutator, hypocenter | C22×C5 | C4×C9 |
| Fitting, socle | C22×C3×C5 | C3×C4 |
| prime radical, cosocle | C22×C3×D5 | C2×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C3 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 90 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 393);