Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of #4175 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| direct product | C3×#4175 |
| Fitting, Sylow | C3×AΣL(1,𝔽27) ⋊ C2 |
| derived | C33 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #4175 |
| commutator, cosocle, hypocenter | C33 | C2×C32 |
| Frattini | C32 | C32×D3 |
| Fitting | C3×AΣL(1,𝔽27) | C2 |
| socle | C32 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C3 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 12 points:
(4,5,7)(6,9,11,10,8,12);(1,2,3)(5,8,9)39 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#4174⋊C2, D3×#4175, #4175⋊D3, … |
| maximal normal subgroup of | C3×#4174⋊C2, D3×#4175, #4175⋊D3, … |
| maximal quotient of | C3×#66322, C2×C3×#4175, C3×#92159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 116);