Finite Group DatabasebetaFinite Group Databasebeta| order | 198 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽11) (order 660) · Inn(G) = D11 · Out(G) = C2×C3×C5 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×D11 |
| unambiguous semidirect product | C11⋊(C2×C9) |
| Fitting, Hall | C9×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | D11 |
| commutator, hypocenter | C11 | C2×C9 |
| Frattini | C3 | C3×D11 |
| Fitting | C9×C11 | C2 |
| socle, cosocle | C3×C11 | C2×C3 |
| prime radical | C3×D11 | C3 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C9 | |
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:
(1,2,4,3,5,7,6,8,9)(11,12)(13,14)(15,16)(17,18)(19,20);(10,11,13,15,17,19,20,18,16,14,12)63 conjugacy classes — view the full character table.
| maximal subgroup of | D9×D11, C2×C9×D11, D11×C27, … |
| maximal normal subgroup of | D9×D11, C2×C9×D11, D11×C27, … |
| maximal quotient of | C9×Dic11, C2×C9×D11, D11×C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(198, 2);