Finite Group DatabasebetaFinite Group Databasebeta| order | 19068 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊Dic227 |
| unambiguous semidirect product | C227⋊(ΓL(1,𝔽8)⋊C4) |
| unambiguous semidirect product | Dic1589⋊C3 |
| derived, Hall | C7×C227 ⋊ C3×C4 |
| Hall | Dic1589 ⋊ C3 |
| Hall | ΓL(1,𝔽8)×C227 ⋊ C4 |
| Sylow | C227 ⋊ ΓL(1,𝔽8)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ Dic227 |
| Sylow | C7 ⋊ C3×Dic227 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽8)⋊D227 |
| commutator, hypocenter | C7×C227 | C3×C4 |
| Fitting, socle, cosocle | C2×C7×C227 | C2×C3 |
| prime radical | C2×ΓL(1,𝔽8)×C227 | C2 |
| LCS | G ▷ C7×C227 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 6356 |
Polycyclic presentation:
2,3,7,227,2|a^2=e,c^a=c^6,d^a=d^226,c^b=c^21144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(19068, 12);