Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of S5 |
| automorphism | Aut(G) = C2×S5 (order 240) · Inn(G) = S5 · Out(G) = C2 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×S5 |
| unambiguous semidirect product | A5⋊(C2×C3) |
| socle | C3×A5 ⋊ C2 |
| derived | A5 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3 | S5 |
| commutator, cosocle, hypocenter | A5 | C2×C3 |
| Frattini | C1 | G |
| socle | C3×A5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C3 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 15 |
A minimal faithful permutation representation — 2 generators on 8 points:
(4,8);(1,2,3)(4,5,6,7,8)21 conjugacy classes — view the full character table.
| maximal subgroup of | D3×S5, C2×C3×S5, C9×S5, … |
| maximal normal subgroup of | D3×S5, C2×C3×S5, C9×S5, … |
| maximal quotient of | C3×#6728, C3×#6729, C3×A5⋊C4, … |
| outer automorphism group of | C22⋊(C22⋊C9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 119);