Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×#272 (order 648) · Inn(G) = D3 · Out(G) = C2×#272 |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×D3×C9 |
| unambiguous semidirect product | C3⋊(C2×C3×C9) |
| Fitting, Sylow | C32×C9 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C9 | D3 |
| commutator, hypocenter | C3 | C2×C3×C9 |
| Frattini | C3 | C32×D3 |
| Fitting | C32×C9 | C2 |
| socle | C33 | C2×C3 |
| prime radical | C32×D3 | C3 |
| cosocle | C32 | C2×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C9 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 15 points:
(5,6)(7,8,10,9,11,13,12,14,15);(1,2,3)(4,5,6)81 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D9, D32×C9, D3×C3⋊D9, … |
| maximal normal subgroup of | C3×D3×D9, D32×C9, D3×C3⋊D9, … |
| maximal quotient of | C3×C9×Dic3, C2×C3×D3×C9, C3×C9×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 33);