Finite Group DatabasebetaFinite Group Databasebeta| order | 36 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = C2×AGL(1,ℤ/9) (order 108) · Inn(G) = D9 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = #72213202 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D18 |
| direct product | C2×D9 |
| unambiguous semidirect product | C9⋊C22 |
| unambiguous semidirect product | (C2×C9)⋊C2 |
| Fitting | C2×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9 |
| commutator, cosocle, hypocenter | C9 | C22 |
| Frattini | C3 | C2×D3 |
| Fitting | C2×C9 | C2 |
| socle | C2×C3 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2 | |
15 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2)(3,4,5,6,7,8,9,10,11);(4,11)(5,10)(6,9)(7,8)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 2 | 2 | 2 | 2 | 9 | 1 | 2 | 2 | 2 | 2 | 9 |
| order | 1 | 3 | 9 | 9 | 9 | 2 | 2 | 6 | 18 | 18 | 18 | 2 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 |
| χ3 | 2 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | -1 | -1 | -1 | 0 |
| χ4 | 2 | -1 | ζ+ζ8 | ζ2+ζ7 | ζ4+ζ5 | 0 | 2 | -1 | ζ+ζ8 | ζ2+ζ7 | ζ4+ζ5 | 0 |
| χ5 | 2 | -1 | ζ2+ζ7 | ζ4+ζ5 | ζ+ζ8 | 0 | 2 | -1 | ζ2+ζ7 | ζ4+ζ5 | ζ+ζ8 | 0 |
| χ6 | 2 | -1 | ζ4+ζ5 | ζ+ζ8 | ζ2+ζ7 | 0 | 2 | -1 | ζ4+ζ5 | ζ+ζ8 | ζ2+ζ7 | 0 |
| χ7 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 |
| χ8 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 |
| χ9 | 2 | 2 | -1 | -1 | -1 | 0 | -2 | -2 | 1 | 1 | 1 | 0 |
| χ10 | 2 | -1 | ζ+ζ8 | ζ2+ζ7 | ζ4+ζ5 | 0 | -2 | 1 | −ζ−ζ8 | −ζ2−ζ7 | −ζ4−ζ5 | 0 |
| χ11 | 2 | -1 | ζ2+ζ7 | ζ4+ζ5 | ζ+ζ8 | 0 | -2 | 1 | −ζ2−ζ7 | −ζ4−ζ5 | −ζ−ζ8 | 0 |
| χ12 | 2 | -1 | ζ4+ζ5 | ζ+ζ8 | ζ2+ζ7 | 0 | -2 | 1 | −ζ4−ζ5 | −ζ−ζ8 | −ζ2−ζ7 | 0 |
Here ζ is a primitive root of unity of order 9 (ζ9 = 1).
| maximal subgroup of | C4×D9, D36, #611, … |
| maximal normal subgroup of | C4×D9, D36, #611, … |
| maximal quotient of | Dic18, C4×D9, D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36, 4);