Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×C9 (order 108) · Inn(G) = D3 · Out(G) = C2×C9 |
| holomorph | G ⋊ Aut(G) = D32×AGL(1,ℤ/27) |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C1 |
| direct product | D3×C27 |
| unambiguous semidirect product | C3⋊(C2×C27) |
| Fitting, Sylow | C3×C27 ⋊ C2 |
| derived | C3 ⋊ C2×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C27 | D3 |
| commutator, hypocenter | C3 | C2×C27 |
| Frattini | C9 | C3×D3 |
| Fitting | C3×C27 | C2 |
| socle | C32 | C2×C9 |
| prime radical | C3×D3 | C9 |
| cosocle | C3×C9 | C2×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C27 | |
19 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 2 generators on 30 points:
(2,3)(4,5,8,6,9,14,11,16,21,7,10,15,12,17,22,19,24,27,13,18,23,20,25,28,26,29,30);(1,2,3)81 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D27, C2×D3×C27, D9×C27, … |
| maximal normal subgroup of | D3×D27, C2×D3×C27, D3×C81, … |
| maximal quotient of | Dic3×C27, C2×D3×C27, D9×C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 8);