Finite Group DatabasebetaFinite Group Databasebeta| order | 396 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C22×C5×AGL(1,ℤ/9) (order 1080) · Inn(G) = D9 · Out(G) = C22×C3×C5 |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2 |
| direct product | C2×C11×D9 |
| unambiguous semidirect product | C9⋊(C22×C11) |
| unambiguous semidirect product | (C2×C9)⋊(C2×C11) |
| Fitting | C2×C9×C11 ⋊ C2 |
| derived, Sylow | C9 ⋊ C22×C11 |
| Hall | C9×C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | D9 |
| commutator, cosocle, hypocenter | C9 | C22×C11 |
| Frattini | C3 | C2×D3×C11 |
| Fitting | C2×C9×C11 | C2 |
| socle | C2×C3×C11 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C11 | |
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)(4,6)(5,9)(7,10)(8,11)(12,13,14,15,16,17,18,19,20,21,22);(1,2)(3,8,6,5,10,7,9,4,11)132 conjugacy classes — view the full character table.
| maximal subgroup of | D9×Dic11, #11716178, #11716180, … |
| maximal normal subgroup of | D9×Dic11, #11716178, #11716180, … |
| maximal quotient of | C11×Dic18, C4×C11×D9, C11×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 8);