Finite Group DatabasebetaFinite Group Databasebeta| order | 36 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = AΓL(1,𝔽9) (order 144) · Inn(G) = G · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72213315 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C32 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32 | C4 |
| prime radical, cosocle | C3⋊D3 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 6 |
| transitive degree | μt(G) = 6 |
A minimal faithful permutation representation — 2 generators on 6 points:
(1,2)(3,5,6,4);(1,3,6)(2,5,4)Polycyclic presentation:
2^2,3^2|a^2=b,c^a=cd^2,d^a=c^2d^2,c^b=c^2,d^b=d^2| C1 | C2 | C3 | C4 | C5 | C6 | |
|---|---|---|---|---|---|---|
| size | 1 | 4 | 4 | 9 | 9 | 9 |
| order | 1 | 3 | 3 | 2 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | 1 | 1 | -1 | ||
| χ4 | 1 | 1 | 1 | -1 | ||
| χ5 | 4 | 1 | -2 | 0 | 0 | 0 |
| χ6 | 4 | -2 | 1 | 0 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36, 9);