Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C4×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = D9 · Out(G) = C3×C4×D4 |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C5×Dic9 |
| unambiguous semidirect product | (C2×C9)⋊(C4×C5) |
| derived, Sylow | C9 ⋊ C2×C4×C5 |
| Hall | C5×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | D9 |
| commutator, hypocenter | C9 | C2×C4×C5 |
| Frattini | C2×C3 | C2×C5×D3 |
| Fitting | C22×C5×C9 | C2 |
| socle, prime radical | C22×C3×C5 | D3 |
| cosocle | C2×C9 | C22×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C22×C5 | |
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)(7,8,9,10,11)(13,15)(14,18)(16,17)(19,20);(1,2)(12,20,15,18,17,16,14,13,19)120 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×C5×C9⋊C8, C5×#3667, C4×C5×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 23);