Finite Group DatabasebetaFinite Group Databasebeta| order | 198 |
| isoclinism | isoclinic class of D33 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽11) (order 1320) · Inn(G) = D33 · Out(G) = C22×C5 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×D33 |
| unambiguous semidirect product | C3⋊(C3×D11) |
| unambiguous semidirect product | C11⋊(C3×D3) |
| Fitting, socle, Hall | C32×C11 ⋊ C2 |
| derived | C3×C11 ⋊ C2×C3 |
| Sylow | C11 ⋊ C3×D3 |
| Sylow | C32 ⋊ D11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D33 |
| commutator, cosocle, hypocenter | C3×C11 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C11 | C2 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 ◁ C3 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 66 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3)(5,6)(8,9)(10,11)(12,13)(14,15)(16,17);(4,5,6)(7,8,10,12,14,16,17,15,13,11,9)54 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D11, D3×D33, (C3⋊D3)⋊D11, … |
| maximal normal subgroup of | C3×D3×D11, D3×D33, (C3⋊D3)⋊D11, … |
| maximal quotient of | C3×Dic33, C2×C3×D33, C3×D99, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(198, 7);