Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of AGL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C7 · Schur multiplier (H2) = C23 |
| direct product | C23×C3×AGL(1,𝔽8) |
| Fitting, socle, Hall | C26×C3 ⋊ C7 |
| derived | C23 ⋊ C23×C3×C7 |
| Sylow | C26 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23 | C23×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C26×C3 | C7 |
| LCS | G ▷ C23 | |
| UCS | C1 ◁ C23×C3 | |
926 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
192 conjugacy classes — view the full character table.
| maximal subgroup of | C3×AGL(1,𝔽8)2, C2×C3×C22⋊AΓL(1,𝔽8), C23×D3×AGL(1,𝔽8) |
| maximal normal subgroup of | C3×AGL(1,𝔽8)2, C2×C3×C22⋊AΓL(1,𝔽8), C23×D3×AGL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11697);