Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C34 |
| unambiguous semidirect product | C3⋊(C3⋊((C3⋊D3)⋊C3)) |
| Fitting, Sylow | C32×H(3,𝔽3) ⋊ C2 |
| derived | C34 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C34 | C2×C3 |
| Frattini | C3 | C3×C3⋊(C3⋊D3) |
| Fitting | C32×H(3,𝔽3) | C2 |
| socle | C33 | C3×D3 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
456 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)(8,13,15,10,11,14)(9,12);(4,5,6);(1,2,3);(7,8,10)(9,11,14)(12,15,13)Polycyclic presentation:
2,3^5|c^a=c^2,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, 242);