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) = C4×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×Dic9 |
| unambiguous semidirect product | C9⋊(C4×C11) |
| derived, Sylow | C9 ⋊ C4×C11 |
| Hall | C9×C11 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | D9 |
| commutator, hypocenter | C9 | C4×C11 |
| Frattini | C2×C3 | D3×C11 |
| Fitting | C2×C9×C11 | C2 |
| socle, prime radical | C2×C3×C11 | D3 |
| cosocle | C2×C9 | C2×C11 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C11 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 396 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,4)(3,7)(5,6)(8,9)(10,13,12,11)(14,15,16,17,18,19,20,21,22,23,24);(1,9,4,7,6,5,3,2,8)132 conjugacy classes — view the full character table.
| maximal subgroup of | #11716175, D99⋊C4, D11×Dic9, … |
| maximal normal subgroup of | #11716175, D99⋊C4, D11×Dic9, … |
| maximal quotient of | C11×C9⋊C8, C2×C11×Dic9, C11×Dic27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(396, 1);