Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89829⋊C3 |
| Fitting | C7×#124 ⋊ C2×C3 |
| Hall | #89829 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #463 |
| Sylow | C7 ⋊ C3×#463 |
248 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,2^3,7|c^2=g,f^2=g,b^a=bf,c^a=cf,f^a=fg,h^a=h^6,c^b=ce,f^b=fg,f^c=fg,h^d=h^271 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1248);