Finite Group DatabasebetaFinite Group Databasebeta| order | 819 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×C13⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
| direct product | ΓL(1,𝔽8)×C13⋊C3 |
| unambiguous semidirect product | (C13×ΓL(1,𝔽8))⋊C3 |
| unambiguous semidirect product | (C7×C13⋊C3)⋊C3 |
| unambiguous semidirect product | #63270⋊C3 |
| unambiguous semidirect product | #63271⋊C3 |
| Fitting, socle, derived, Hall | C7×C13 ⋊ C32 |
| Sylow | C13 ⋊ C3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3×C13⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C13 | C32 |
| LCS | G ▷ C7×C13 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 91 |
35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(819, 6);