Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of AΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C23×AΓL(1,𝔽8) |
| unambiguous semidirect product | AGL(1,𝔽8)⋊(C23×C3) |
| unambiguous semidirect product | (C2×AGL(1,𝔽8))⋊(C22×C3) |
| Fitting, socle, Sylow | C26 ⋊ ΓL(1,𝔽8) |
| derived | AGL(1,𝔽8) ⋊ C23×C3 |
| Hall | C23×AGL(1,𝔽8) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | AΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | AGL(1,𝔽8) | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C26 | ΓL(1,𝔽8) |
| LCS | G ▷ AGL(1,𝔽8) | |
| UCS | C1 ◁ C23 | |
431 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×AΓL(1,𝔽8), AGL(1,𝔽8)×AΓL(1,𝔽8) |
| maximal normal subgroup of | C2×A4×AΓL(1,𝔽8), AGL(1,𝔽8)×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11688);