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) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C5×D9 |
| unambiguous semidirect product | C9⋊(C22×C5) |
| unambiguous semidirect product | (C2×C9)⋊(C2×C5) |
| Fitting | C2×C5×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C22×C5 |
| Hall | C5×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D9 |
| commutator, cosocle, hypocenter | C9 | C22×C5 |
| Frattini | C3 | C2×C5×D3 |
| Fitting | C2×C5×C9 | C2 |
| socle | C2×C3×C5 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C5 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 90 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,4,5,6,7)(9,11)(10,14)(12,15)(13,16);(1,2)(8,13,11,10,15,12,14,9,16)60 conjugacy classes — view the full character table.
| maximal subgroup of | D9×Dic5, #67080, #67082, … |
| maximal normal subgroup of | D9×Dic5, #67080, #67082, … |
| maximal quotient of | C5×Dic18, C4×C5×D9, C5×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(180, 10);