Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of C3⋊D9 |
| automorphism | Aut(G) = C2×#72210336 (order 5832) · Inn(G) = C3⋊D9 · Out(G) = C2×#272 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| direct product | C3×C3⋊D9 |
| unambiguous semidirect product | C3⋊(C3×D9) |
| Fitting, Sylow | C32×C9 ⋊ C2 |
| derived | C3×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊D9 |
| commutator, cosocle, hypocenter | C3×C9 | C2×C3 |
| Frattini | C3 | C3×C3⋊D3 |
| Fitting | C32×C9 | C2 |
| socle | C33 | D3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C3 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,6)(8,10)(9,13)(11,12)(14,15);(4,5,6);(7,15,10,13,12,11,9,8,14)45 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D9, D3×C3⋊D9, #66431, … |
| maximal normal subgroup of | C3×D3×D9, D3×C3⋊D9, #66431, … |
| maximal quotient of | C3×C3⋊Dic9, C2×C3×C3⋊D9, C9×C3⋊D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 38);