Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/9) (order 216) · Inn(G) = D9 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = D3×#72213202 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D9 |
| Fitting | C2×C3×C9 ⋊ C2 |
| derived | C9 ⋊ C22×C3 |
| Sylow | C3×C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D9 |
| commutator, cosocle, hypocenter | C9 | C22×C3 |
| Frattini | C3 | C2×C3×D3 |
| Fitting | C2×C3×C9 | C2 |
| socle | C2×C32 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C3 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,4,5)(7,9)(8,12)(10,13)(11,14);(1,2)(6,11,9,8,13,10,12,7,14)36 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×D9, #6192, #6194, … |
| maximal normal subgroup of | Dic3×D9, #6192, #6194, … |
| maximal quotient of | C3×Dic18, C3×C4×D9, C3×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 23);