Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = C2×D9 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D36 |
| unambiguous semidirect product | C4⋊(C3×D9) |
| Fitting | C3×C4×C9 ⋊ C2 |
| Sylow | C3×C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D9 |
| commutator, cosocle | C2×C9 | C22×C3 |
| Frattini | C2×C3 | C2×C3×D3 |
| Fitting | C3×C4×C9 | C2 |
| socle | C2×C32 | C2×D3 |
| hypercenter | C3×C4 | D9 |
| hypocenter | C9 | C3×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C3 ◁ 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)(4,9,11,10,5,7,6,12,8)(13,16,15,14);(5,7)(6,10)(8,9)(11,12)(14,16)63 conjugacy classes — view the full character table.
| maximal subgroup of | #88415, #88417, #88421, … |
| maximal normal subgroup of | #88415, #88417, #88421, … |
| maximal quotient of | C3×Dic36, C3×#3664, C3×D72, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 46);