Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×C3×D4×AGL(1,𝔽5) (order 960) · Inn(G) = D5 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×Dic5 |
| derived, Sylow | C5 ⋊ C2×C4×C9 |
| Hall | C5×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | D5 |
| commutator, hypocenter | C5 | C2×C4×C9 |
| Frattini | C2×C3 | C2×C3×D5 |
| Fitting | C22×C5×C9 | C2 |
| socle, prime radical | C22×C3×C5 | C2×C3 |
| cosocle | C2×C3×C5 | C22×C3 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22×C9 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4,5,6)(8,9)(10,11)(12,19,18,14,13,20,17,16,15);(1,2)(7,8,10,11,9)144 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×Dic9, #10623557, Dic9⋊Dic5, … |
| maximal normal subgroup of | Dic5×Dic9, #10623557, Dic9⋊Dic5, … |
| maximal quotient of | C2×C9×#168, C9×#681, C4×C9×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 18);