Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×D3×D4 (order 288) · Inn(G) = D3 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C9×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C4×C9) |
| derived | C3 ⋊ C2×C4×C9 |
| Sylow | C3×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | D3 |
| commutator, hypocenter | C3 | C2×C4×C9 |
| Frattini | C2×C3 | C2×C3×D3 |
| Fitting | C22×C3×C9 | C2 |
| socle, prime radical | C22×C32 | C2×C3 |
| cosocle | C2×C32 | C22×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C9 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(4,5)(6,7,8,9)(10,17,16,12,11,18,15,14,13);(1,2)(3,4,5)108 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×Dic9, Dic9⋊Dic3, #88442, … |
| maximal normal subgroup of | Dic3×Dic9, Dic9⋊Dic3, #88442, … |
| maximal quotient of | C2×C9×C3⋊C8, C9×#208, C4×C9×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 56);