Finite Group DatabasebetaFinite Group Databasebeta| order | 66 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = D11 · Out(G) = C2×C5 |
| holomorph | G ⋊ Aut(G) = D3×#11925892⋊C2 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×D11 |
| unambiguous semidirect product | C11⋊(C2×C3) |
| Fitting, socle, Hall | C3×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D11 |
| commutator, cosocle, hypocenter | C11 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C3×C11 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C3 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 33 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3)(5,6)(7,8)(9,10)(11,12)(13,14);(4,5,7,9,11,13,14,12,10,8,6)21 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D11, C2×C3×D11, C9×D11, … |
| maximal normal subgroup of | D3×D11, C2×C3×D11, C9×D11, … |
| maximal quotient of | C3×Dic11, C2×C3×D11, C9×D11, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(66, 2);