Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽5) (order 240) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×AGL(1,𝔽5) |
| unambiguous semidirect product | D5⋊(C4×C9) |
| Fitting | C2×C5×C9 ⋊ C4 |
| derived, Sylow | C5 ⋊ C2×C4×C9 |
| Hall | C5×C9 ⋊ C2×C4 |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 16 points:
(4,5,6,7)(8,15,14,10,9,16,13,12,11);(1,2)(3,4,5,7,6)90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 43);