Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×A5⋊C4 |
| unambiguous semidirect product | A5⋊(C3×C4) |
| derived | A5 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3 | S5 |
| commutator, hypocenter | A5 | C3×C4 |
| Frattini | C2 | C3×S5 |
| socle, prime radical | C2×C3×A5 | C2 |
| cosocle | C2×A5 | C2×C3 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3 | |
75 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
42 conjugacy classes — view the full character table.
| maximal subgroup of | #13088282, ΓL(2,𝔽4)⋊C4, D3×A5⋊C4, … |
| maximal normal subgroup of | #13088282, ΓL(2,𝔽4)⋊C4, D3×A5⋊C4, … |
| maximal quotient of | C3×A5⋊C8, C3×GL(2,𝔽5), C3×#91143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 412);