Finite Group DatabasebetaFinite Group Databasebeta| order | 132 |
| 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) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D66 |
| direct product | C2×D33 |
| unambiguous semidirect product | (C2×C3)⋊D11 |
| unambiguous semidirect product | (C2×C11)⋊D3 |
| Fitting, socle | C2×C3×C11 ⋊ C2 |
| derived, Hall | C3×C11 ⋊ C22 |
| Sylow | C11 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D33 |
| commutator, cosocle, hypocenter | C3×C11 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C11 | C2 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 ◁ C2 | |
20 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)(6,7,8,9,10,11,12,13,14,15,16);(4,5)(7,16)(8,15)(9,14)(10,13)(11,12)36 conjugacy classes — view the full character table.
| maximal subgroup of | D33⋊C4, #63141, #63142, … |
| maximal normal subgroup of | D33⋊C4, #63141, #63142, … |
| maximal quotient of | Dic66, C4×D33, D132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(132, 9);