Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of S5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C3×C5×S5 |
| unambiguous semidirect product | A5⋊(C2×C3×C5) |
| socle | C3×C5×A5 ⋊ C2 |
| derived | A5 ⋊ C2×C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3×C5 | S5 |
| commutator, cosocle, hypocenter | A5 | C2×C3×C5 |
| Frattini | C1 | G |
| socle | C3×C5×A5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C3×C5 | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 75 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×C5×S5 |
| maximal normal subgroup of | C2×C3×C5×S5 |
| maximal quotient of | C3×C5×A5⋊C4, C2×C3×C5×S5 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 559);