Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C23×AGL(1,ℤ/9) (order 432) · Inn(G) = D9 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×C4×D9 |
| Fitting | C3×C4×C9 ⋊ C2 |
| derived | C9 ⋊ C2×C3×C4 |
| Sylow | C3×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D9 |
| commutator, hypocenter | C9 | C2×C3×C4 |
| Frattini | C2×C3 | C2×C3×D3 |
| Fitting | C3×C4×C9 | C2 |
| socle | C2×C32 | C2×D3 |
| prime radical | C2×C3×D9 | C2 |
| cosocle | C2×C9 | C22×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C3×C4 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3)(5,7)(6,10)(8,11)(9,12);(4,9,7,6,11,8,10,5,12)(13,16,15,14)72 conjugacy classes — view the full character table.
| maximal subgroup of | D9×C3⋊C8, #88413, C3×C8×D9, … |
| maximal normal subgroup of | D9×C3⋊C8, #88413, C3×C8×D9, … |
| maximal quotient of | C3×C8×D9, C3×#3663, C3×C4×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 45);