Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of C3⋊D9 |
| automorphism | Aut(G) = D4×#72210336 (order 23328) · Inn(G) = C3⋊D9 · Out(G) = D4×#272 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C3⋊Dic9 |
| unambiguous semidirect product | (C2×C3)⋊Dic9 |
| unambiguous semidirect product | (C2×C9)⋊Dic3 |
| derived, Sylow | C3×C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C2×C4 |
| Frattini | C2×C3 | C2×C3⋊D3 |
| Fitting | C22×C3×C9 | C2 |
| socle, prime radical | C22×C32 | D3 |
| cosocle | C2×C3×C9 | C22 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C22 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(4,5)(6,7,8,9)(11,13)(12,16)(14,15)(17,18);(1,2)(3,4,5);(10,18,13,16,15,14,12,11,17)60 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×C3⋊(C9⋊C8), #88532, C4×C3⋊Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 69);