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) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic9 |
| unambiguous semidirect product | C9⋊C4 |
| derived, Sylow | C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9 |
| commutator, hypocenter | C9 | C4 |
| Frattini, socle, prime radical | C2×C3 | D3 |
| Fitting, cosocle | C2×C9 | C2 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2 | |
9 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(3,4)(5,6,7,8,9,10,11,12,13);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)Polycyclic presentation:
2^2,3^2|a^2=b,c^3=d^2,c^a=c^2d,d^a=d^2| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | C11 | C12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 9 | 9 |
| order | 1 | 2 | 3 | 6 | 9 | 9 | 9 | 18 | 18 | 18 | 4 | 4 |
| 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 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | ||
| χ4 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | ||
| χ5 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| χ6 | 2 | 2 | -1 | -1 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | 0 | 0 |
| χ7 | 2 | 2 | -1 | -1 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | 0 | 0 |
| χ8 | 2 | 2 | -1 | -1 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | 0 | 0 |
| χ9 | 2 | -2 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 0 | 0 |
| χ10 | 2 | -2 | -1 | 1 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | −ζ22+ζ26−ζ34 | ζ14+ζ22 | −ζ14−ζ26+ζ34 | 0 | 0 |
| χ11 | 2 | -2 | -1 | 1 | −ζ14−ζ22 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | ζ14+ζ22 | −ζ14−ζ26+ζ34 | −ζ22+ζ26−ζ34 | 0 | 0 |
| χ12 | 2 | -2 | -1 | 1 | ζ14+ζ26−ζ34 | ζ22−ζ26+ζ34 | −ζ14−ζ22 | −ζ14−ζ26+ζ34 | −ζ22+ζ26−ζ34 | ζ14+ζ22 | 0 | 0 |
Here ζ is a primitive root of unity of order 36 (ζ36 = 1); i = ζ9.
| maximal subgroup of | Dic18, C4×D9, C2×Dic9, … |
| maximal normal subgroup of | Dic18, C4×D9, C2×Dic9, … |
| maximal quotient of | C9⋊C8, C2×Dic9, Dic27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36, 1);