Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of A4×A5 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| direct product | A4×A5 |
| unambiguous semidirect product | C22⋊(C3×A5) |
| unambiguous semidirect product | (C22×A5)⋊C3 |
| Fitting | C22 ⋊ C3×A5 |
| socle, derived | C22×A5 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, hypocenter | C22×A5 | C3 |
| Fitting, cosocle | C22 | C3×A5 |
| LCS | G ▷ C22×A5 | |
| UCS | C1 | |
63 conjugacy classes of subgroups.
| transitive degree | μt(G) = 20 |
20 conjugacy classes — view the full character table.
| derived subgroup of | A4⋊S5, S4×A5, C2×S4×A5, … |
| maximal subgroup of | A4×S5, A4⋊S5, S4×A5, … |
| maximal normal subgroup of | A4×S5, A4⋊S5, S4×A5, … |
| maximal quotient of | SL(2,𝔽3)×A5, SL(2,𝔽5)⋊A4, A4×SL(2,𝔽5), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 768);