Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C2×S4×AGL(1,ℤ/9) (order 2592) · Inn(G) = D9 · Out(G) = C2×C3×S4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×D9 |
| Fitting | C22×C3×C9 ⋊ C2 |
| derived | C9 ⋊ C23×C3 |
| Sylow | C3×C9 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D9 |
| commutator, cosocle, hypocenter | C9 | C23×C3 |
| Frattini | C3 | C22×C3×D3 |
| Fitting | C22×C3×C9 | C2 |
| socle | C22×C32 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C22×C3 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,4)(5,6,7)(9,11)(10,14)(12,15)(13,16);(3,4)(8,13,11,10,15,12,14,9,16);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D9)⋊Dic3, C3×(C2×D9)⋊C4, C2×Dic3×D9, … |
| maximal normal subgroup of | (C2×D9)⋊Dic3, C3×(C2×D9)⋊C4, C2×Dic3×D9, … |
| maximal quotient of | C2×C3×Dic18, C2×C3×C4×D9, C2×C3×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 108);