Finite Group DatabasebetaFinite Group Databasebeta| order | 330 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽11) |
| automorphism | Aut(G) = D3×AGL(1,𝔽11) (order 660) · Inn(G) = G · Out(G) = C2 |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊AGL(1,𝔽11) |
| unambiguous semidirect product | (C11⋊C5)⋊D3 |
| unambiguous semidirect product | D33⋊C5 |
| Fitting, socle, derived, Hall | C3×C11 ⋊ C2×C5 |
| Hall | C3×C11⋊C5 ⋊ C2 |
| Hall | D33 ⋊ C5 |
| Hall | C11⋊C5 ⋊ D3 |
| Sylow | C11 ⋊ C5×D3 |
| Sylow | C3 ⋊ AGL(1,𝔽11) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C11 | C2×C5 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 33 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(5,9,10,8,13,6,7,14,11,12);(1,2,3)(4,5,8,13,7,10,14,9,12,11,6)Polycyclic presentation:
2,5,3,11|c^a=c^2,d^a=d^10,d^b=d^418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(330, 3);