Finite Group DatabasebetaFinite Group Databasebeta| order | 198 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C2×C5×AGL(1,ℤ/9) (order 540) · Inn(G) = D9 · Out(G) = C2×C3×C5 |
| homology | Abelianization (H1) = C2×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×D9 |
| unambiguous semidirect product | C9⋊(C2×C11) |
| Fitting, Hall | C9×C11 ⋊ C2 |
| derived, Sylow | C9 ⋊ C2×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | D9 |
| commutator, cosocle, hypocenter | C9 | C2×C11 |
| Frattini | C3 | D3×C11 |
| Fitting | C9×C11 | C2 |
| socle | C3×C11 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C11 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 99 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(3,7)(5,8)(6,9)(10,11,12,13,14,15,16,17,18,19,20);(1,6,4,3,8,5,7,2,9)66 conjugacy classes — view the full character table.
| maximal subgroup of | D9×D11, C2×C11×D9, C11×D27, … |
| maximal normal subgroup of | D9×D11, C2×C11×D9, C3×C11×D9, … |
| maximal quotient of | C11×Dic9, C2×C11×D9, C11×D27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(198, 1);