Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C23×AGL(1,ℤ/9) (order 432) · Inn(G) = C2×D9 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#611 |
| Fitting | C22×C3×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 | C22×C3×C9 | C2 |
| socle | C2×C32 | C2×D3 |
| hypercenter | C22×C3 | D9 |
| hypocenter | C9 | C3×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3)(4,9,11,10,5,7,6,12,8)(14,16);(5,7)(6,10)(8,9)(11,12)(13,14)(15,16)63 conjugacy classes — view the full character table.
| maximal subgroup of | #88658, #88663, D3×#611, … |
| maximal normal subgroup of | #88658, #88663, D3×#611, … |
| maximal quotient of | C3×Dic9⋊C4, C3×(C2×D9)⋊C4, C3×#3672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 57);