Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of S4×AGL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C2 |
| direct product | S4×AGL(1,𝔽8) |
| unambiguous semidirect product | A4⋊(C2×AGL(1,𝔽8)) |
| unambiguous semidirect product | (C23×S4)⋊C7 |
| unambiguous semidirect product | (A4×AGL(1,𝔽8))⋊C2 |
| Fitting, socle | C25 ⋊ D3×C7 |
| derived | C23×A4 ⋊ C2×C7 |
| Hall | C23×S4 ⋊ C7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C23×A4 | C2×C7 |
| Fitting, socle | C25 | D3×C7 |
| LCS | G ▷ C23×A4 | |
| UCS | C1 | |
141 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | S4×AΓL(1,𝔽8) |
| maximal normal subgroup of | S4×AΓL(1,𝔽8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11687);