Finite Group DatabasebetaFinite Group Databasebeta| order | 180 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C2×C4×AGL(1,ℤ/9) (order 432) · Inn(G) = D9 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic9 |
| unambiguous semidirect product | C9⋊(C4×C5) |
| derived, Sylow | C9 ⋊ C4×C5 |
| Hall | C5×C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D9 |
| commutator, hypocenter | C9 | C4×C5 |
| Frattini | C2×C3 | C5×D3 |
| Fitting | C2×C5×C9 | C2 |
| socle, prime radical | C2×C3×C5 | D3 |
| cosocle | C2×C9 | C2×C5 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C5 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12,13,14,15,16,17,18);(6,8,7,9)(11,18)(12,17)(13,16)(14,15)60 conjugacy classes — view the full character table.
| maximal subgroup of | #67077, #67079, D5×Dic9, … |
| maximal normal subgroup of | #67077, #67079, D5×Dic9, … |
| maximal quotient of | C5×C9⋊C8, C2×C5×Dic9, C5×Dic27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 1);