Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of C3⋊(C3⋊AGL(1,ℤ/9)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊AGL(1,ℤ/9)) |
| Fitting, Sylow | C32×3−1+2 ⋊ C2 |
| derived | C32×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×C9 | C2×C3 |
| Frattini | C3 | C3×C3⋊(C3⋊D3) |
| Fitting | C32×3−1+2 | C2 |
| socle | C33 | C3×D3 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 15 points:
(2,3)(5,6)(8,12,15,10,11,14)(9,13);(4,5,6);(1,2,3);(7,15,10,13,11,12,9,8,14)Polycyclic presentation:
2,3^5|c^3=f^2,c^a=c^2f,d^a=d^2,e^a=e^2,f^a=f^2,c^b=cf54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 244);