Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | Aut(G) = C9×((C3⋊D3)⋊C3)⋊C2 (order 972) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C2×C9 |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊D3)⋊C27 |
| Fitting, Sylow | C32⋊C27 ⋊ C2 |
| derived | C32 ⋊ C2×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C2×C27 |
| Frattini | C3×C9 | C3×D3 |
| Fitting | C32⋊C27 | C2 |
| socle | C32 | D3×C9 |
| prime radical | C3×C3⋊D3 | C9 |
| cosocle | C32×C9 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C9 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 36 points:
(2,7,9,4,5,8)(3,6)(10,15,27,18,30,29,32,11,20,13,23,33,26,35,14,17,16,28,19,31,36,12,22,21,25,24,34);(1,2,4)(3,5,8)(6,9,7)Polycyclic presentation:
2,3^5|b^3=c,c^3=d,e^a=e^2,f^a=f^2,e^b=ef90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 16);