Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C22×C4×AGL(1,ℤ/9) (order 864) · Inn(G) = D9 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C5×C9⋊C8 |
| unambiguous semidirect product | C9⋊(C5×C8) |
| derived, Sylow | C9 ⋊ C5×C8 |
| Hall | C5×C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | D9 |
| commutator, hypocenter | C9 | C5×C8 |
| Frattini | C3×C4 | C5×D3 |
| Fitting | C4×C5×C9 | C2 |
| socle, prime radical | C2×C3×C5 | Dic3 |
| cosocle | C4×C9 | C2×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C4×C5 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,4)(3,7)(5,6)(8,9)(10,15,12,17,13,11,16,14)(18,19,20,21,22);(1,9,4,7,6,5,3,2,8)120 conjugacy classes — view the full character table.
| maximal subgroup of | #10623539, #10623541, D5×C9⋊C8, … |
| maximal normal subgroup of | #10623539, #10623541, D5×C9⋊C8, … |
| maximal quotient of | C5×C9⋊C16, C2×C5×C9⋊C8, C5×C27⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 1);