Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C9 · Schur multiplier (H2) = C2 |
| direct product | C4×C5×C22⋊C9 |
| unambiguous semidirect product | (C22×C4)⋊(C5×C9) |
| unambiguous semidirect product | (C22×C4×C5)⋊C9 |
| derived | C22 ⋊ C4×C5×C9 |
| Hall | C22×C4×C5 ⋊ C9 |
| Sylow | C22×C4 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4×C5 | A4 |
| commutator, hypocenter | C22 | C4×C5×C9 |
| Frattini | C2×C3 | C2×C5×A4 |
| Fitting | C22×C3×C4×C5 | C3 |
| socle, prime radical | C23×C3×C5 | C2×C3 |
| cosocle | C23×C3 | C2×C3×C5 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C3×C4×C5 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 180 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C5×(C22⋊C9)⋊C8, (C22⋊C9)⋊#168, C22⋊C9×#168, … |
| maximal normal subgroup of | C5×(C22⋊C9)⋊C8, (C22⋊C9)⋊#168, C22⋊C9×#168, … |
| maximal quotient of | C4×C5×Q8⋊C9, C5×C8×C22⋊C9, C5×#131⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 152);