Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×SL(2,ℤ/4) |
| unambiguous semidirect product | A4⋊(C3×C4×C5) |
| derived | A4 ⋊ C3×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | S4 |
| commutator, hypocenter | A4 | C3×C4×C5 |
| Frattini | C2 | C3×C5×S4 |
| Fitting, socle | C23×C3×C5 | D3 |
| prime radical | C2×C3×C5×A4 | C2 |
| cosocle | C2×A4 | C2×C3×C5 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C3×C5 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 180 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#91899, C3×#91901, C3×D5×SL(2,ℤ/4), … |
| maximal normal subgroup of | C3×#91899, C3×#91901, C3×D5×SL(2,ℤ/4), … |
| maximal quotient of | C3×C5×A4⋊C8, C3×C5×#859, C3×C5×GU(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 528);