Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽11) (order 1320) · Inn(G) = D11 · Out(G) = C22×C3×C5 |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×D11 |
| unambiguous semidirect product | C11⋊(C22×C9) |
| unambiguous semidirect product | (C2×C11)⋊(C2×C9) |
| Fitting | C2×C9×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C22×C9 |
| Hall | C9×C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | D11 |
| commutator, hypocenter | C11 | C22×C9 |
| Frattini | C3 | C2×C3×D11 |
| Fitting | C2×C9×C11 | C2 |
| socle | C2×C3×C11 | C2×C3 |
| prime radical | C2×C3×D11 | C3 |
| cosocle | C3×C11 | C22×C3 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C9 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 198 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,7,11,5,10,6,8,4,9)(13,14)(15,16)(17,18)(19,20)(21,22);(1,2)(12,13,15,17,19,21,22,20,18,16,14)126 conjugacy classes — view the full character table.
| maximal subgroup of | D11×Dic9, #11716180, #11716181, … |
| maximal normal subgroup of | D11×Dic9, #11716180, #11716181, … |
| maximal quotient of | C9×Dic22, C4×C9×D11, C9×D44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 7);