Finite Group DatabasebetaFinite Group Databasebeta| order | 198 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = GL(2,𝔽3)×AGL(1,𝔽11) (order 5280) · Inn(G) = D11 · Out(G) = C5×GL(2,𝔽3) |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C32×D11 |
| unambiguous semidirect product | C11⋊(C2×C32) |
| Fitting, socle, Hall | C32×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | D11 |
| commutator, cosocle, hypocenter | C11 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C32×C11 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C32 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 99 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,5,6)(8,9)(10,11)(12,13)(14,15)(16,17);(1,2,3)(7,8,10,12,14,16,17,15,13,11,9)63 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D11, C3⋊D3×D11, C2×C32×D11, … |
| maximal normal subgroup of | C3×D3×D11, C3⋊D3×D11, C2×C32×D11, … |
| maximal quotient of | C32×Dic11, C2×C32×D11, C3×C9×D11, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(198, 5);