Finite Group DatabasebetaFinite Group Databasebeta| order | 342 |
| isoclinism | isoclinic class of D19⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×D19⋊C3 |
| unambiguous semidirect product | D19⋊C32 |
| unambiguous semidirect product | (C19⋊C3)⋊(C2×C3) |
| unambiguous semidirect product | (C3×D19)⋊C3 |
| Fitting, socle | C3×C19 ⋊ C2×C3 |
| derived, Sylow | C19 ⋊ C2×C32 |
| Hall | C3×C19⋊C3 ⋊ C2 |
| Hall | D19 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D19⋊C3 |
| commutator, cosocle, hypocenter | C19 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C19 | C2×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C3 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 57 |
A minimal faithful permutation representation — 2 generators on 22 points:
(5,9,10,6,7,11)(8,17,14,12,13,19)(15,20,22,21,18,16);(1,2,3)(4,5,8,14,20,17,21,7,11,16,22,10,9,15,13,18,19,12,6)27 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D19⋊C3, C2×C3×D19⋊C3, C9×D19⋊C3, … |
| maximal normal subgroup of | D3×D19⋊C3, C2×C3×D19⋊C3, C9×D19⋊C3, … |
| maximal quotient of | C3×(C19⋊C3)⋊C4, C2×C3×D19⋊C3, C9×D19⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(342, 9);