Finite Group DatabasebetaFinite Group Databasebeta| order | 342 |
| isoclinism | isoclinic class of AGL(1,𝔽19) |
| automorphism | Aut(G) = AGL(1,𝔽19) (order 342) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,𝔽19) |
| unambiguous semidirect product | #4296⋊C2 |
| Fitting, socle, derived, Sylow | C19 ⋊ C2×C9 |
| Hall | #4296 ⋊ C2 |
| Hall | D19 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C19 | C2×C9 |
| prime radical | D19⋊C3 | C3 |
| cosocle | C19⋊C3 | C2×C3 |
| LCS | G ▷ C19 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 19 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,13,14,7,16,12,10,6,15,3,9,19,5,17,8,11,4,18);(1,2,6,13,8,16,15,10,7,14,19,5,11,18,17,12,9,4,3)Polycyclic presentation:
2,3^2,19|b^3=c,d^a=d^18,d^b=d^17,d^c=d^1119 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(342, 7);