Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of A5 |
| automorphism | Aut(G) = C2×S5 (order 240) · Inn(G) = A5 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×A5 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2×C3 | A5 |
| commutator, hypocenter | A5 | C2×C3 |
| Frattini | C1 | G |
| LCS | G ▷ A5 | |
| UCS | C1 ◁ C2×C3 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 30 |
A minimal faithful permutation representation — 2 generators on 10 points:
(7,8,9);(1,2)(3,4,5)(6,7,8,9,10)30 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 122);