Finite Group DatabasebetaFinite Group Databasebeta| order | 132 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C22×AGL(1,𝔽11) (order 440) · Inn(G) = D11 · Out(G) = C22×C5 |
| holomorph | G ⋊ Aut(G) = D3×#72213693 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic11 |
| unambiguous semidirect product | C11⋊(C3×C4) |
| derived, Sylow | C11 ⋊ C3×C4 |
| Hall | C3×C11 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D11 |
| commutator, hypocenter | C11 | C3×C4 |
| Frattini | C2 | C3×D11 |
| Fitting, socle, prime radical | C2×C3×C11 | C2 |
| cosocle | C2×C11 | C2×C3 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 132 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3)(4,5)(6,7)(8,9,10,11,12,13,14,15,16,17,18);(4,6,5,7)(9,18)(10,17)(11,16)(12,15)(13,14)42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic11, D33⋊C4, #63142, … |
| maximal normal subgroup of | D3×Dic11, D33⋊C4, #63142, … |
| maximal quotient of | C3×C11⋊C8, C2×C3×Dic11, C9×Dic11, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(132, 2);