Finite Group DatabasebetaFinite Group Databasebeta| order | 342 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×C9 (order 216) · Inn(G) = D3 · Out(G) = C22×C9 |
| homology | Abelianization (H1) = C2×C3×C19 · Schur multiplier (H2) = C1 |
| direct product | C3×D3×C19 |
| unambiguous semidirect product | C3⋊(C2×C3×C19) |
| Fitting, socle, Hall | C32×C19 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C19 |
| Sylow | C32 ⋊ C2×C19 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C19 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C3×C19 |
| Frattini | C1 | G |
| Fitting, socle | C32×C19 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C19 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 114 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3)(5,6)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25);(4,5,6)171 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D19, D32×C19, D3×D57, … |
| maximal normal subgroup of | C3×D3×D19, D32×C19, D3×D57, … |
| maximal quotient of | C3×Dic3×C19, C2×C3×D3×C19, C3×D9×C19, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(342, 14);