Finite Group DatabasebetaFinite Group Databasebeta| order | 330 |
| isoclinism | isoclinic class of C11⋊C5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = C11⋊C5 · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C2×C3×C11⋊C5 |
| unambiguous semidirect product | (C2×C11)⋊(C3×C5) |
| unambiguous semidirect product | (C2×C3×C11)⋊C5 |
| Fitting, socle, Hall | C2×C3×C11 ⋊ C5 |
| derived, Sylow | C11 ⋊ C2×C3×C5 |
| Hall | C3×C11 ⋊ C2×C5 |
| Hall | C2×C11 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C11⋊C5 |
| commutator, cosocle, hypocenter | C11 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C11 | C5 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C3 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 66 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,4,5)(7,8,10,13,12)(9,11,15,16,14);(6,7,9,12,8,10,14,16,11,13,15)42 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic11⋊C5, Dic3×C11⋊C5, C3⋊(Dic11⋊C5), … |
| maximal normal subgroup of | C3×Dic11⋊C5, Dic3×C11⋊C5, C3⋊(Dic11⋊C5), … |
| maximal quotient of | C3×C4×C11⋊C5, C22×C3×C11⋊C5, C2×C9×C11⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(330, 4);